text:
By ThisisRnB on May 05, 2014
Let's be honest, getting anything unreleased from Michael Jackson is like finding gold. While we are still consistently playing the brand new single "Love Never Felt So Good," from the posthumous album Xscape, another one of the projects much-anticipated songs has surfaced online. Although it sounds like it is the original "demo" version and not the newly christened "contemporized" version, but many will say they prefer hearing it like this.
With only a week to go before the album hits stores, Epic is giving fans an early preview with a snippet of a new song each day this week. Today's release is "Chicago," which is also known as "She Was Lovin' Me," a track that didn't make the cut on Jackson's 2001 album Invincible. According to Sony, Jackson fell in love with the song, which was written by Cory Rooney, the first time he heard it in 1999.
Get your first listen of MJ's "Chicago" below:
Tags: Michael Jackson,xscape
New Music: Usher - Good Kisser
New Music: Nelly - Thanks To My Ex
mreagan88	8 years ago
Uh oh this sounds really good im impressed
olu33	8 years ago
I like this,
waratties	8 years ago
hiphop_hype


Monica wrote for Mashable's Tech section with a focus on retail, internet of things, and the intersections of technology and social justice. She holds a degree in creative writing from Brown University, and has previously written for Dow Jones Media, the New York Post, Yahoo Finance, and others. In her free time, she can be found attempting to cook Asian food, buying board games, and looking for new hobbies.


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\section{Introduction}
In \cite{Rieffel1} M. A. Rieffel introduced a C*-algebraic framework for deformation quantization whereby a C*-algebra $A$ equipped with an action of $\mb{R}^{n}$ by automorphisms and further supplied with a skew-symmetric $J\in M_{n}(\mathbb{R})$, produces a C*-algebra $A_{J}$ with multiplication $\times_{J}$,
often referred to as the \textit{Rieffel deformation} of the original algebra. Several other well-known examples of C*-algebras can be shown to arise in this way.
The K-theory of the deformed algebra was studied in \cite{Rieffel3}, revealing that the deformed algebra $A_{J}$ and the original algebra $A$ have the same K-groups. There, the key technique was to show that $A_{J}$ was strongly Morita equivalent to a certain crossed product of (a stabilization and suspension of) $A$ by $\mb{R}^{n}$, followed by an application of the Connes-Thom result in K-theory, stability and Morita invariance of the K-functor.

Some operator algebraic approaches to deformation quantization use various notions of ``twists'', utilizing an action (e.g. of a group) combined with a distinguished element satisfying some cocyclicity-condition (e.g. a group 2-cocycle) as ingredients towards deforming a given algebra equipped with said action. One such procedure is explored by Kasprzak in \cite{kasprzak} where a locally compact abelian group $G$ acts on a C*-algebra $A$. Given a 2-cocycle $\psi$ on the dual group $\widehat{G}$, there is a method for obtaining a deformed algebra $A^{\psi}$. This procedure encompasses in particular Rieffel deformation as the case $G=\mb{R}^{n}$ with a certain choice of 2-cocycle on $\widehat{\mb{R}^{n}}$. Concerning K-theory, there is an isomorphism $A^{\psi}\rtimes G\cong A\rtimes G$ of crossed products which, for the case $G = \mb{R}^{n}$, when combined with the Connes-Thom result yields an identification of the K-groups of the deformed and undeformed algebras respectively.

The present paper discusses the continuous field over $[0,1]$ of the Rieffel deformation and shows that the evaluation map is a KK-equivalence. Namely, for a C*-algebra $A$ with an action of $\mb{R}^{n}$ and given a skew-symmetric matrix $J$, taking $t\in [0,1]$ and using $tJ$ as the skew-symmetric matrix gives the Rieffel deformation $A_{tJ}$. This will constitute a continuous field $(A_{tJ})_{t\in [0,1]}$ as was already explored in the original monograph \cite{Rieffel1}. We show that the evaluation map of the bundle algebra $\pi_{h}: \Gamma( (A_{tJ})_{t\in [0,1]} )\longrightarrow A_{hJ}$, for each $h\in [0,1]$, is a KK-equivalence. To accomplish this we shall employ the deformation approach of Kasprzak and consider a deformed bundle algebra $B^{\psi}$ which will be a $C([0,1])$-algebra equipped with a fibrewise action. As such, RKK-theory naturally enters and we show that $B^{\psi}$ is an \textit{RKK-fibration} in the sense of \cite{Echterhoff-Nest-Oyono-2} by appealing to the fibrewise action and the Connes-Thom result in RKK-theory. The important consequence of being an RKK-fibration here, is that the evaluation map of the $C([0,1])$-algebra becomes a KK-equivalence. Finally, the deformed bundle algebra $B^{\psi}$ will be shown to be $C([0,1])$-linearly *-isomorphic to the bundle algebra  $\Gamma( (A_{tJ})_{t\in [0,1]} )$ of the continuous field of the Rieffel deformation, thus yielding the promised result.

We now give a more specific outline of the paper. Section $\ref{KasprzakApproach}$ explains the approach to deformation taken in \cite{kasprzak}, where one starts with the action of a locally compact abelian group $G$ with a 2-cocycle $\psi$ on the Pontryagin dual $\widehat{G}$. A certain subalgebra $A^{\psi}\subseteq M(A\rtimes G)$ is obtained as the Landstad algebra of the G-product $(A\rtimes G, \lambda, \widehat{\alpha})$. After presenting the basic preliminaries and some of the needed results, we specialize to $G=\mb{R}^{n}$ with our specific 2-cocycle $\psi_{J}$. Section $\ref{sectionBundleAndRKK}$ discusses the relevant bundle and collects a few needed ingredients from \cite{Echterhoff-Nest-Oyono-2} on RKK-fibrations and their relation to KK-equivalences, and then proceeds to establish that the aforementioned bundle is an RKK-fibration. Section $\ref{RieffelField}$ recalls the main notions of Rieffel deformation, the associated continuous field and the relation to the 2-cocycle deformation. The main result regarding the evaluation map of the bundle algebra of the continuous field is then achieved as a consequence of the RKK-fibration laid forth in the preceding section.
In section 5 we comment on the special case called theta deformation, in which the action is not by $\mb{R}^{n}$ but $\mathbb{T}^{n}$. There, a different bundle algebra is plausible. Namely, taking a fix-point algebra description of the deformed algebra, we use a strong Morita equivalence to a certain crossed product algebra by the integers and work with an integer crossed product bundle algebra. One is able to show that the related bundle evaluation map has a KK-contractible kernel by applying the Pimsner-Voiculescu six-term exact sequence, so the KK-equivalence follows. Finally we describe the invariance of the index pairing which we understand as a KK-product between elements of the K-group with (in particular) the Fredholm module coming from a spectral triple.

\section{Twisting by a 2-cocycle}\label{KasprzakApproach}
We recall the approach to deformation as in \cite{kasprzak}. The idea is based on twisting a dual C*-dynamical system by a 2-cocycle of the dual group. First we recollect some preliminaries on C*-dynamical systems and $G$-products (cf. \cite{pedersen} \S 7.8).

\begin{definition} Let $G$ be a locally compact abelian group and $\widehat{G}$ its Pontryagin dual group. Let $B$ be a C*-algebra with a strict-continuous unitary-valued homomorphism $\lambda: G\longrightarrow M(B)$, and let $\widehat{\rho}$ be a strongly continuous action $\widehat{\rho}:\widehat{G}\longrightarrow Aut(B)$ satisfying
\[ \widehat{\rho}_{\chi}(\lambda_{\gamma}) = \chi(\gamma) \lambda_{\gamma} \]
for all $\chi\in \widehat{G}$ and $\gamma\in G$.
The triple $(B,\lambda,\widehat{\rho})$ is called a \textit{G-product}. One also simply refers to $B$ as a $G$-product when the rest is implicitly understood.
\end{definition}

Given a $G$-product $(B, \lambda,\widehat{\rho})$, one may extend the given unitary representation $\lambda$ to the *-homomorphism $\lambda: C^{*}(G)\longrightarrow M(B)$. Using the Fourier transform to identify $C^{*}(G)\cong C_{0}(\widehat{G})$ we write $\lambda: C_{0}(\widehat{G})\longrightarrow M(B)$. This map is injective and we often omit $\lambda$ from the notation.

\begin{definition} Let $(B, \lambda,\widehat{\rho})$ be a $G$-product and let $x\in M(B)$. The element $x$ satisfies the \textit{Landstad conditions} if:
\begin{align*}
\mbox{(i)}\,\,& \widehat{\rho}_{\chi}(x) = x \mbox{ for all } \chi\in\widehat{G},\\
\mbox{(ii)}\,\,& \mbox{the map } G\ni\gamma \mapsto \lambda_{\gamma}x\lambda_{\gamma}^{*}\in M(B) \mbox{ is norm continuous},\\
\mbox{(iii)}\,\,& fxg\in B \mbox{ for all } f,g\in C_{0}(\widehat{G}).
\end{align*}
The set of elements satisfying the Landstad conditions turns out to be a subalgebra in $M(B)$. We shall refer to this subalgebra as the \textit{Landstad algebra} of the $G$-product.
\end{definition}

The foremost example of a $G$-product is produced by the crossed product construction. Indeed, given an abelian C*-dynamical system $(B,G,\alpha)$, the triple $(B\rtimes_{\alpha}G, \lambda, \widehat{\alpha})$ is a $G$-product whose Landstad algebra is precisely $B$. The following result states that any $G$-product arises in this way.

\begin{theorem}\label{theo1}\cite[Theorem 7.8.8]{pedersen} A C*-algebra $B$ is a $G$-product $(B,\lambda,\widehat{\rho})$ if and only if there exists a C*-dynamical system $(C,G,\beta)$ for which $B \cong C\rtimes_{\beta}G$. The C*-dynamical system is unique up to covariant isomorphism, the C*-algebra $C$ is just the associated Landstad algebra and $\beta = Ad\,\lambda$.
\end{theorem}

Recall that a \textit{2-cocycle} $\psi$ on the abelian group $\widehat{G}$ is a continuous function
\[ \psi: \widehat{G}\times\widehat{G}\longrightarrow \mathbb{T} \]
satisfying
\renewcommand{\labelenumi}{(\roman{enumi})}
\begin{enumerate}
\item $\psi(e,\chi) = \psi(\chi,e) = 1$ for all $\chi\in\widehat{G}$,
\item $\psi(\chi_{1}, \chi_{2}+\chi_{3})\psi(\chi_{2},\chi_{3}) = \psi(\chi_{1}+\chi_{2}, \chi_{3})\psi(\chi_{1},\chi_{2})$ for all $\chi_{1},\chi_{2},\chi_{3}\in\widehat{G}$
\end{enumerate}

Given an element $\chi\in\widehat{G}$, define the function $\psi_{\chi} \in C_{b}(\widehat{G})$ by
\[ \psi_{\chi}(\widehat{\sigma}) = \psi(\chi,\widehat{\sigma}) \mbox{ for } \widehat{\sigma}\in\widehat{G}. \]
Observing that $C_{b}(\widehat{G}) = M(C_{0}(\widehat{G}))$, use the obvious extension $\lambda: C_{b}(\widehat{G}) \longrightarrow M(B)$ and obtain unitaries
\begin{equation}\label{unitaries}
U_{\chi} = \lambda(\psi_{\chi}) \in M(B).
\end{equation}
The 2-cocycle condition for $\psi$ implies the following commutation rule for these unitaries
\[ U_{\chi_{1}+\chi_{2}} = \bar{\psi}(\chi_{1},\chi_{2})U_{\chi_{1}}\widehat{\rho}_{\chi_{1}}(U_{\chi_{2}}) .\]

\begin{lemma}\label{lemma1}\cite[Theorem 3.1]{kasprzak} Let $(B,\lambda,\widehat{\rho})$ be a $G$-product and $\psi$ a 2-cocycle on $\widehat{G}$. Use the unitaries of $(\ref{unitaries})$ to define the strongly continuous action $\widehat{\rho}^{\psi}: \widehat{G}\longrightarrow Aut(B)$,
\[ \widehat{\rho}^{\psi}_{\chi}(b) = U_{\chi}^{*}\widehat{\rho}_{\chi}(b)U_{\chi} \]
for $\chi\in\widehat{G}$ and $b\in B$. Then $(B, \lambda, \widehat{\rho}^{\psi})$ is a $G$-product.
\end{lemma}

\begin{definition}[Kasprzak deformation]  Let $A$ be a separable C*-algebra with strongly continuous action $\alpha: G\longrightarrow Aut(A)$ of the locally compact abelian group $G$, and $\psi$ a 2-cocycle on $\widehat{G}$. The $G$-product $(A\rtimes_{\alpha}G,\lambda,\widehat{\alpha})$ gives rise to the $G$-product $(A\rtimes_{\alpha}G,\lambda,\widehat{\alpha}^{\psi})$ by Lemma $\ref{lemma1}$. The deformed algebra $A^{\psi}$ is by definition the Landstad algebra of the $G$-product $(A\rtimes_{\alpha}G,\lambda,\widehat{\alpha}^{\psi})$.
\end{definition}

An interesting result is obtained by considering the original action on the deformed algebra. Denote by $\alpha^{\psi}: G\longrightarrow Aut(A^{\psi})$ the action $\alpha_{g}^{\psi}(x) = \lambda_{g}x\lambda_{g}^{*}$. If we consider the crossed product of the C*-dynamical system $(A^{\psi}, G, \alpha^{\psi})$ we get
\begin{lemma}\label{KasprzakLandstadTheorem} $A^{\psi}\rtimes_{\alpha^{\psi}}G \cong A\rtimes_{\alpha}G$.
\end{lemma}
\begin{proof} The proof is a literal application of Theorem $\ref{theo1}$. Indeed, let $B = A\rtimes_{\alpha}G$ and consider the $G$-product $(B, \lambda, \widehat{\alpha}^{\psi}) = (A\rtimes_{\alpha}G, \lambda, \widehat{\alpha}^{\psi})$. The Landstad algebra of this $G$-product is what we have called $A^{\psi}$ by definition, which is the algebra $C = A^{\psi}$ referred to in Theorem $\ref{theo1}$. Furthermore $\alpha^{\psi} = Ad\,\lambda$, which is the action $\beta$ in that theorem. In other words the C*-dynamical system is $(C,G,\beta) = (A^{\psi}, G, \alpha^{\psi})$ and the theorem yields the isomorphism $B \cong C\rtimes_{\beta}G$, in our case $A\rtimes_{\alpha}G \cong A^{\psi}\rtimes_{\alpha^{\psi}}G$ as claimed.

Following \cite{pedersen} and \cite{Landstad}, we may further describe the *-isomorphism $A^{\psi}\rtimes_{\alpha^{\psi}}G \longrightarrow A\rtimes_{\alpha}G$ as mapping $y\otimes g \mapsto y\lambda_{g}$, for $y\in A^{\psi}$ and $g\in C_{c}(G)$.
\end{proof}

Let our separable C*-algebra $A$ be equipped with a strongly continuous action $\sigma:\mb{R}^{n}\longrightarrow Aut(A)$, and let $J\in M_{n}(\mathbb{R})$ be a skew-symmetric matrix. On $\mb{R}^{n}$ we consider the symmetric bicharacter
\[ e: \mb{R}^{n}\times\mb{R}^{n}\longrightarrow\mathbb{T} \]
\[ e(u,v) = e^{2\pi i\, u\cdot v} \]
which gives the group isomorphism $\mb{R}^{n} \cong \widehat{\mb{R}^{n}}$ by $u\mapsto e^{1}_{u}$ where $e^{1}_{u}(v) = e(u,v)$.
We use the 2-cocycle $\psi_{J}: \widehat{\mb{R}^{n}}\times\widehat{\mb{R}^{n}}\longrightarrow \mathbb{T}$,
\begin{equation}\label{2cocycleEq}
\psi_{J}(e^{1}_{u},e^{1}_{v}) = e^{1}_{u}(J v) = e(u, Jv) = e^{2\pi i\, u\cdot Jv}.
\end{equation}
By Lemma $\ref{lemma1}$ the $\mb{R}^{n}$-product $(A\rtimes_{\sigma}\mb{R}^{n}, \lambda, \widehat{\sigma})$ combined with the 2-cocycle $\psi_{J}$ gives the $\mb{R}^{n}$-product $(A\rtimes_{\sigma}\mb{R}^{n}, \lambda, \widehat{\sigma}^{\psi_{J}})$, and the deformed algebra $A^{\psi_{J}}$ is the corresponding Landstad algebra.

\section{Bundle structure and RKK-fibration}\label{sectionBundleAndRKK}
Let $X$ be a locally compact Hausdorff space. A C*-algebra $B$ is called a $C_{0}(X)$-algebra (cf. \cite[1.5]{Kasparov1}) if there is a non-degenerate *-homomorphism $\Phi_{B}: C_{0}(X) \longrightarrow ZM(B)$. One also writes $fb = \Phi_{B}(f)b$, for $f\in C_{0}(X)$ and $b\in B$. For each $x\in X$, letting $I_{x} = \{f\in C_{0}(X):\, f(x) = 0\}$ be the ideal of functions vanishing at $x$, then $I_{x}B\subseteq B$ is an ideal and the quotient $B_{x} = B/(I_{x}B)$ is called the \textit{fiber} over $x$. The quotient map $q_{x}:B\longrightarrow B_{x}$ is also referred to as \textit{evaluation at} $x$.

Recall that we are considering a strongly continuous action $\sigma: \mb{R}^{n}\longrightarrow Aut(A)$ on a separable C*-algebra $A$, and a real skew-symmetric matrix $J$.
Let $B = C([0,1])\otimes A = C([0,1],A)$ be equipped with the obvious $C([0,1])$-algebra structure $\Phi_{B}: C([0,1])\longrightarrow ZM(B)$, $\Phi_{B}(f)(g\otimes a) = fg\otimes a$. Define the action $\beta:\mb{R}^{n}\longrightarrow Aut(B)$
\begin{equation}\label{actionBeta}
\beta_{x}(y)(s) = \sigma_{\sqrt{s}x}(y(s)),
\end{equation}
for $x\in\mb{R}^{n}$, $y\in B$, $s\in [0,1]$. Let $\psi = \psi_{J}$ be the 2-cocycle from $(\ref{2cocycleEq})$. Then the 2-cocycle deformation $B^{\psi}$ is by definition the Landstad algebra of the $\mb{R}^{n}$-product $(B\rtimes_{\beta}\mb{R}^{n}, \lambda, \widehat{\beta}^{\psi})$. Recall the action $\beta^{\psi}:\mb{R}^{n}\longrightarrow Aut(B^{\psi})$ from the remark preceding Lemma $\ref{KasprzakLandstadTheorem}$.

\begin{lemma} The deformed algebra $B^{\psi}$ is a $C([0,1])$-algebra and the action $\beta^{\psi}:\mb{R}^{n}\longrightarrow Aut(B^{\psi})$ is fiberwise. There is a $C([0,1])$-linear *-isomorphism
\[ B^{\psi}\rtimes_{\beta^{\psi}}\mb{R}^{n} \longrightarrow B\rtimes_{\beta}\mb{R}^{n}. \]
\end{lemma}
\begin{proof} Clearly the action $\beta$ on $B$ is $C([0,1])$-linear, i.e. for every $x\in\mb{R}^{n}$, $\beta_{x}(\Phi_{B}(f)y) = \Phi_{B}(f)\beta_{x}(y)$ for every $f\in C([0,1])$ and $y\in B$. This entails that $\Phi_{B\rtimes_{\beta}\mb{R}^{n}}: C([0,1])\longrightarrow ZM(B\rtimes_{\beta}\mb{R}^{n})$ given by $(\Phi_{B\rtimes_{\beta}\mb{R}^{n}}(f)y)(v) = \Phi_{B}(f)(y(v))$ gives a $C([0,1])$-algebra structure on $B\rtimes_{\beta}\mb{R}^{n}$. Concerning the dual action $\widehat{\beta}:\widehat{\mb{R}^{n}}\longrightarrow Aut(B\rtimes_{\beta}\mb{R}^{n})$, for each $w\in\widehat{\mb{R}^{n}}$ the canonically extended automorphism $\widehat{\beta}_{w}: M(B\rtimes_{\beta}\mb{R}^{n}) \longrightarrow M(B\rtimes_{\beta}\mb{R}^{n})$ satisfies $\widehat{\beta}_{w}(\Phi_{B\rtimes_{\beta}\mb{R}^{n}}(f)) = \Phi_{B\rtimes_{\beta}\mb{R}^{n}}(f)$ for every $f\in C([0,1])$. It then follows
that for any $y\in M(B\rtimes_{\beta}\mb{R}^{n})$,
\begin{align*}
\widehat{\beta}^{\psi}_{w}(\Phi_{B\rtimes_{\beta}\mb{R}^{n}}(f)) &= U_{w}^{*}\widehat{\beta}_{w}(\Phi_{B\rtimes_{\beta}\mb{R}^{n}}(f))U_{w} = U_{w}^{*}\Phi_{B\rtimes_{\beta}\mb{R}^{n}}(f)U_{w} \\
&= \Phi_{B\rtimes_{\beta}\mb{R}^{n}}(f)U^{*}_{w}U_{w} = \Phi_{B\rtimes_{\beta}\mb{R}^{n}}(f),
\end{align*}
i.e. $\Phi_{B\rtimes_{\beta}\mb{R}^{n}}( C([0,1]) ) \subseteq M(B\rtimes_{\beta}\mb{R}^{n})^{\widehat{\beta}^{\psi}} = B^{\psi}$. Combined with the fact that $\Phi_{B\rtimes_{\beta}\mb{R}^{n}}( C([0,1]) ) \subseteq ZM(B\rtimes_{\beta}\mb{R}^{n})$, this entails that we may define $\Phi_{B^{\psi}} = \Phi_{B\rtimes_{\beta}\mb{R}^{n}}$ to obtain a $C([0,1])$-algebra structure on $B^{\psi}$.

The action $\beta^{\psi}:\mb{R}^{n}\longrightarrow Aut(B^{\psi})$ is $\beta^{\psi}_{x}(y) = \lambda_{x}y\lambda_{x}^{*}$, for $y\in B^{\psi}$, and so
\begin{align*}
\beta_{x}^{\psi}(\Phi_{B^{\psi}}(f)y) &= \beta_{x}^{\psi}(\Phi_{B\rtimes_{\beta}\mb{R}^{n}}(f)y) = \lambda_{x}(\Phi_{B\rtimes_{\beta}\mb{R}^{n}}(f)y)\lambda_{x}^{*} \\
&= \Phi_{B\rtimes_{\beta}\mb{R}^{n}}(f)\lambda_{x}y\lambda_{x}^{*} =  \Phi_{B^{\psi}}(f)\beta_{x}^{\psi}(y),
\end{align*}
i.e. the action $\beta^{\psi}$ is fiberwise and hence naturally makes the crossed product $B^{\psi}\rtimes_{\beta^{\psi}}\mb{R}^{n}$ a $C([0,1])$-algebra where $\Phi_{B^{\psi} \rtimes_{\beta^{\psi}}\mb{R}^{n}}:C([0,1])\longrightarrow ZM(B^{\psi} \rtimes_{\beta^{\psi}}\mb{R}^{n})$ is given by the composition of $\Phi_{B^{\psi}}$ with the inclusion $M(B^{\psi})\subseteq M(B^{\psi} \rtimes_{\beta^{\psi}}\mb{R}^{n})$. By Lemma $\ref{KasprzakLandstadTheorem}$
\begin{equation}\label{thirdProp}
B^{\psi} \rtimes_{\beta^{\psi}}\mb{R}^{n} \cong B\rtimes_{\beta}\mb{R}^{n},
\end{equation}
and we claim this *-isomorphism to be $C([0,1])$-linear. Indeed, denote this *-isomorphism $S: B^{\psi} \rtimes_{\beta^{\psi}}\mb{R}^{n} \longrightarrow B\rtimes_{\beta}\mb{R}^{n}$, which by Lemma $\ref{KasprzakLandstadTheorem}$ can be described as $S(y\otimes g) = y\lambda_{g}$ for $y\in B^{\psi}$ and $g\in C_{c}(\mb{R}^{n})$, and it follows that
\begin{align*}
S(\Phi_{B^{\psi}\rtimes_{\beta^{\psi}}\mb{R}^{n}}(f)(y\otimes g)) &= S(\Phi_{B^{\psi}}(f)y\otimes g) \\
&= \Phi_{B^{\psi}}(f)y\lambda_{g} \\
&= \Phi_{B\rtimes_{\beta}\mb{R}^{n}}(f)y\lambda_{g} \\
&= \Phi_{B\rtimes_{\beta}\mb{R}^{n}}(f)S(y\otimes g)
\end{align*}
 for any $f\in C([0,1])$, i.e. $S\circ \Phi_{B^{\psi}\rtimes_{\beta^{\psi}}\mb{R}^{n}} = \Phi_{B\rtimes_{\beta}\mb{R}^{n}}\circ S$.
\end{proof}

Let $f: Y\longrightarrow X$ be a continuous map between locally compact spaces. The pullback construction gives a $C_{0}(X)$-algebra structure on $C_{0}(Y)$, since $f^{*}: C_{0}(X) \longrightarrow C_{b}(Y)$ and $C_{b}(Y) = ZM(C_{0}(Y))$, we let $\Phi_{C_{0}(Y)}: C_{0}(X) \longrightarrow ZM(C_{0}(Y))$, $\Phi_{C_{0}(Y)}(k) = f^{*}(k)$ be the pointwise multiplication operator by the pullback
\[ \Phi_{C_{0}(Y)}(k)h = f^{*}(k)h \]
for $k\in C_{0}(X)$, $h\in C_{0}(Y)$.

Given a $C_{0}(X)$-algebra $B$, a locally compact space $Y$ and $f:Y\longrightarrow X$ a continuous map, the \textit{pullback} $f^{*}(B)$ of $B$ along $f$ is the $C_{0}(Y)$-algebra
\begin{equation}\label{pullbackDef}
f^{*}(B) = C_{0}(Y)\otimes_{C_{0}(X)}B.
\end{equation}
The balanced tensor product in $(\ref{pullbackDef})$ is by definition the quotient of $C_{0}(Y)\otimes B$ by the ideal generated by
\[ \{ \Phi_{C_{0}(Y)}(k)g\otimes b - g\otimes \Phi_{B}(k)b \, \big| \quad g\in C_{0}(Y), \, b\in B, \, k\in C_{0}(X) \}. \]
The $C_{0}(Y)$-algebra structure on $f^{*}(B)$ is pointwise multiplication on the left, $\Phi_{f^{*}(B)}: C_{0}(Y) \longrightarrow ZM(f^{*}(B))$, $\Phi_{f^{*}(B)}(h)(g\otimes b) = hg\otimes b$, for $h, g\in C_{0}(Y)$ and $b\in B$.
Note that the fiber $f^{*}(B)_{y}$ over $y\in Y$ is $B_{f(y)}$. Indeed, as in the balanced tensor product one has $I_{y}C_{0}(Y)\otimes_{C_{0}(X)}B = C_{0}(Y)\otimes_{C_{0}(X)}I_{f(y)}B$, then
\begin{align*}
 f^{*}(B)_{y} &= C_{0}(Y)\otimes_{C_{0}(X)}B/ I_{y}C_{0}(Y)\otimes_{C_{0}(X)}B \\
&= C_{0}(Y)\otimes_{C_{0}(X)}B/ C_{0}(Y)\otimes_{C_{0}(X)}I_{f(y)}B \\
&= B/ I_{f(y)}B = B_{f(y)}.
\end{align*}

Recall that given two graded, separable C*-algebras $A$ and $B$, the group $KK(A,B)$ is the set of Kasparov $A$-$B$-modules (also called Kasparov cycles) modulo an appropriate equivalence relation (e.g. homotopy equivalence). Briefly, a Kasparov $A$-$B$-module is a triple $(E,\phi,F)$ where $E$ is a countably generated right Hilbert $B$-module, $\phi: A\longrightarrow \mathcal{L}_{B}(E)$ is a *-homomorphism and $F\in \mathcal{L}_{B}(E)$ is a degree 1 operator such that $[F,\phi(a)]$, $(F^{2}-1)\phi(a)$ and $(F-F^{*})\phi(a)$ are elements of $\mathcal{K}_{B}(E)$ for any $a\in A$.

The \textit{KK-product} is a bilinear map
\[ KK(A,D)\times KK(D,B) \longrightarrow KK(A,B) \]
\[ (\textbf{x},\textbf{y}) \mapsto \textbf{x}\textbf{y} \]
where $A$, $B$ and $D$ are separable (and $D$ is $\sigma$-unital) C*-algebras. There is a multiplicatively neutral element $\textbf{1}_{D}= [(D, id, 0)] \in KK(D,D)$ such that for any $\textbf{x}\in KK(A,D)$ and $\textbf{y}\in KK(D,B)$ one has $\textbf{x}\textbf{1}_{D} = \textbf{x}$ and $\textbf{1}_{D}\textbf{y} = \textbf{y}$.

An element $\textbf{x}\in KK(A,B)$ is called a \textit{KK-equivalence} if it is invertible with respect to the KK-product, i.e. if there exists an element $\textbf{y}\in KK(B,A)$ such that $\textbf{xy} = \textbf{1}_{A} \in KK(A,A)$ and $\textbf{yx} = \textbf{1}_{B}\in KK(B,B)$.

Given a graded *-homomorphism $\phi: A\longrightarrow B$, then $(B,\phi,0)$ is the naturally associated Kasparov $A$-$B$-module. We say $\phi$ is a KK-equivalence if the corresponding element $[(B,\phi,0)]\in KK(A,B)$ is a KK-equivalence.

Regarding $C_{0}(X)$-algebras there is a further refinement of the KK-groups called RKK-groups (\cite{Kasparov1}). Namely, for two $C_{0}(X)$-algebras $A$ and $B$, the group $RKK(X; A,B)$ consists of Kasparov $A$-$B$-modules $(E,\phi,F)$ as before, only with the additional requirement
\begin{equation}\label{DefRKKrelation}
(fa)\cdot e\cdot b = a\cdot e\cdot (fb)
\end{equation}
for any $f\in C_{0}(X)$, $a\in A$, $b\in B$ and $e\in E$.

The notions $RKK(X; \cdot, \cdot)$-product and $RKK(X; \cdot, \cdot)$-equivalence are similar to those of the KK-counterpart.

We let $\Delta^{p}\subseteq\mathbb{R}^{p+1}$ denote the standard $p$-simplex.

\begin{definition} A $C_{0}(X)$-algebra $B$ is called a \textit{KK-fibration} if for every positive integer $p$, every continuous map $f:\Delta^{p}\longrightarrow X$ and every element $v\in\Delta^{p}$ the evaluation $q_{v}: f^{*}(B)\longrightarrow B_{f(v)}$ is a KK-equivalence.
\end{definition}

\begin{definition} A $C_{0}(X)$-algebra $B$ is called an \textit{RKK-fibration} if for every positive integer $p$, every continuous map $f:\Delta^{p}\longrightarrow X$ and every element $v\in\Delta^{p}$, $f^{*}(B)$ is $RKK(\Delta^{p}; \cdot,\cdot)$-equivalent to $C(\Delta^{p},B_{f(v)})$.
\end{definition}

\begin{remark}\label{trivialRKKfibration} Given a C*-algebra $A$, the canonical $C_{0}(X)$-algebra $B = C_{0}(X)\otimes A$ is an RKK-fibration. Indeed, given $f: \Delta^{p}\longrightarrow X$ and $v\in\Delta^{p}$, the pullback
\[ f^{*}(B) =  C(\Delta^{p})\otimes_{C_{0}(X)}C_{0}(X)\otimes A \]
is $C(\Delta^{p})$-linearly *-isomorphic to $C(\Delta^{p},B_{f(v)}) = C(\Delta^{p})\otimes B_{f(v)} = C(\Delta^{p})\otimes A$ by the map
\[ h\otimes g\otimes a \mapsto \Phi_{C(\Delta^{p})}(g)h\otimes a = f^{*}(g)h\otimes a, \]
where $h\in C(\Delta^{p})$, $g\in C_{0}(X)$ and $a\in A$.
This implies the required $RKK(\Delta^{p}; \cdot,\cdot)$-equivalence.
\end{remark}
Note also that the property of being an RKK-fibration is preserved under RKK-equivalence.

The following observation (\cite[Remark 1.4]{Echterhoff-Nest-Oyono-2}) will be useful.
\begin{lemma}\label{RKKfibKK} An $RKK$-fibration is a $KK$-fibration.
\end{lemma}
\begin{proof} Suppose $B$ is an $RKK$-fibration, let $f: \Delta^{p}\longrightarrow X$ and $v\in \Delta^{p}$. Concisely put, we get the following comutative diagram in the KK category in which all arrows but the right vertical arrow are already known to be isomorphisms
\[
\begin{CD}
C(\Delta^{p}, B_{f(v)}) @>{\textbf{r}}>> f^{*}(B) \\
@V{ev_{v}}VV               @VV{q_{v}}V \\
B_{f(v)} @>>{r(v)}> B_{f(v)}
\end{CD}
\]
so it follows that the right vertical arrow $q_{v}$ must be an isomorphism as well.

In details, by assumption there exists an invertible element
\[ \textbf{r}\in RKK(\Delta^{p}; C(\Delta^{p},B_{f(v)}), f^{*}(B)). \]
Here $C(\Delta^{p},B_{f(v)}) = C(\Delta^{p})\otimes B_{f(v)}$ is the canonical $C(\Delta^{p})$-algebra with constant fiber $B_{f(v)}$ over each point of $\Delta^{p}$, its bundle projection map being just the evaluation $ev_{w}:C(\Delta^{p},B_{f(v)})\longrightarrow B_{f(v)}$, $ev_{w}(f\otimes b) = f(w)b$, for any $w\in\Delta^{p}$, and it gives in particular the KK-equivalence $[ev_{v}] \in KK(C(\Delta^{p},B_{f(v)}),B_{f(v)})$. Recall also that $f^{*}(B)$ has fiber $B_{f(v)}$ over the point $v\in\Delta^{p}$, denote this bundle projection map $q_{v}$. From the invertible element $\textbf{r}\in RKK(\Delta^{p}; C(\Delta^{p},B_{f(v)}), f^{*}(B))$ we get an invertible element $r(v)\in KK(B_{f(v)}, B_{f(v)})$ which implements the KK-equivalence between the fibers. It follows from
\[ [q_{v}]\cdot\textbf{r} = r(v)[ev_{v}] \]
that $[q_{v}] = r(v)[ev_{v}]\textbf{r}^{-1}$ is a KK-equivalence.
\end{proof}

Recall the Connes-Thom isomorphism in K-theory $K_{i}(A\rtimes_{\alpha}\mathbb{R}) \cong K_{i-1}(A)$, $i=0, 1$, where $\alpha\in Aut(A)$ is a continuous action. The analogous result in KK-theory establishes the existence of an invertible element $\textbf{t}_{\alpha}\in KK^{1}(A,A\rtimes_{\alpha}\mathbb{R}) = KK(SA,A\rtimes_{\alpha}\mathbb{R})$, \textit{the Thom element}. In other words, $A$ and $A\rtimes_{\alpha}\mathbb{R}$ are KK-equivalent with dimension shift $1$.
The case of an $\mb{R}^{n}$-action is handled by repeated application of the above, yielding a KK-equivalence with total dimension shift $n \,(mod\,2)$. In dealing with $C_{0}(X)$-algebras we shall make use of the following RKK-version of the Connes-Thom isomorphism (see \cite[$\S 4$]{Kasparov1})

\begin{theorem}\cite[Theorem 3.5]{Echterhoff-Nest-Oyono-1} \label{ConnesThomKasparov} Let  $A$ be a $C_{0}(X)$-algebra and $\alpha: \mb{R}^{n}\longrightarrow Aut(A)$ a fibrewise action. There exists an invertible element
\[ \textbf{t}_{\alpha} \in RKK^{n}(X; A, A\rtimes_{\alpha}\mb{R}^{n}). \]
Hence $A$ and $A\rtimes_{\alpha}\mb{R}^{n}$ are $RKK$-equivalent with dimension shift $n\,(mod\,2)$.
\end{theorem}

\begin{theorem}\label{theoremRKKfibration} $B^{\psi}$ is an RKK-fibration.
\end{theorem}
\begin{proof} It follows from Theorem $\ref{ConnesThomKasparov}$ that $B^{\psi}$ is $RKK([0,1]; \cdot,\cdot)$-equivalent, with dimension shift $n\,(mod\,2)$, to $B^{\psi}\rtimes_{\beta^{\psi}}\mb{R}^{n}$. By the isomorphism $(\ref{thirdProp})$ the latter algebra is $RKK([0,1]; \cdot,\cdot)$-equivalent to $B\rtimes_{\beta}\mb{R}^{n}$, which by Theorem $\ref{ConnesThomKasparov}$ again is $RKK([0,1]; \cdot,\cdot)$-equivalent, with another dimension shift $n\,(mod\,2)$, to $B = C([0,1])\otimes A$. The total dimension shift thus far is $2n\,(mod\,2) = 0$, i.e. the net effect being no dimension shift, so $B^{\psi}$ is plainly $RKK([0,1]; \cdot,\cdot)$-equivalent to $B$. Finally, the algebra $B = C([0,1])\otimes A$ is clearly an $RKK$-fibration (Remark $\ref{trivialRKKfibration}$), thus proving the claim.
\end{proof}

It follows from Theorem $\ref{theoremRKKfibration}$ and Lemma $\ref{RKKfibKK}$ that $B^{\psi}$ is a KK-fibration. Taking the identity function of the $1$-simplex, $f: \Delta^{1} = [0,1]\longrightarrow [0,1]$, $f(s) = s$, we conclude that the evaluation map $q_{s}: B^{\psi}\longrightarrow (B^{\psi})_{s}$ is a KK-equivalence. Although maybe not completely transparent thus far, it will be made clear in section $\ref{RieffelField}$ that $B^{\psi} \cong \Gamma( (A_{tJ})_{t\in [0,1]} )$ is the bundle algebra of the continuous field over $[0,1]$ of the Rieffel deformation and $(B^{\psi})_{s} \cong A_{sJ}$ is the fiber over the point $s\in [0,1]$.

\section{The continuous field of the Rieffel deformation}\label{RieffelField}
We briefly recall some of the basic facts from \cite{Rieffel1} concerning Rieffel deformation. Let  $\sigma:\mb{R}^{n}\longrightarrow Aut(A)$ be a strongly continuous action on a separable C*-algebra $A$, and $J\in M_{n}(\mathbb{R})$ a skew-symmetric matrix.
Let $\tau$ be the translation action on the Frechet space $C_{b}(\mb{R}^{n},A)$ and let $C_{u}(\mb{R}^{n},A)$ be the largest subspace on which $\tau$ is strongly continuous. Denote by $\mathcal{B}^{A} = \mathcal{B}^{A}(\mb{R}^{n})\subseteq C_{u}(\mb{R}^{n},A)$ the subalgebra of smooth elements for the action $\tau$. For any $F\in \mathcal{B}^{A}(\mb{R}^{n}\times \mb{R}^{n})$ the integral
\[ \iint F(u,v) e^{2\pi i u\cdot v}\, du\,dv \]
exists, as shown in \cite[Chapter 1]{Rieffel1} by considerations of oscillatory integrals. For $f, g\in \mathcal{B}^{A}(\mb{R}^{n})$, the function $(u,v)\mapsto \tau_{Ju}(f)(x)\tau_{v}(g)(x)$ is an element of $\mathcal{B}^{A}(\mb{R}^{n}\times \mb{R}^{n})$ for each $x\in\mb{R}^{n}$, hence the following integral is well defined
\begin{equation}\label{deformedProdFunctions}
(f\times_{J}g)(x) = \iint \tau_{Ju}(f)(x)\tau_{v}(g)(x)e^{2\pi i (u\cdot v)},
\end{equation}
and it turns out $\times_{J}$ defines an associative product on $\mathcal{B}^{A}(\mb{R}^{n})$, and we denote by $\mathcal{B}^{A}_{J} = (\mathcal{B}^{A}(\mb{R}^{n}), \times_{J})$ this algebra structure.
Let $\mathcal{S}^{A}\subseteq \mathcal{B}^{A}$ be the subspace of $A$-valued Schwartz functions. This is naturally a right Hilbert $A$-module for the $A$-valued inner product $\langle f,g\rangle_{A} = \int f(x)^{*}g(x)$. Considering the product $\times_{J}$, it turns out $\mathcal{S}^{A}_{J}$ is an ideal in $\mathcal{B}^{A}_{J}$, this still being compatible with the Hilbert C*-module structure. In this way $\mathcal{S}^{A}_{J}$ carries a representation $L = L^{J}$ of $\mathcal{B}^{A}_{J}$ by adjointable operators
\[ L: \mathcal{B}^{A}_{J} \longrightarrow \mathcal{L}(\mathcal{S}^{A}_{J}) \]
\[ L_{f}(\xi) = f\times_{J}\xi, \quad f\in \mathcal{B}^{A}_{J}, \, \xi\in \mathcal{S}^{A}_{J}. \]

Let $A^{\infty}\subseteq A$ denote the dense *-subalgebra of smooth elements for the action $\sigma$. For $a, b\in A^{\infty}$, the function $(u,v)\mapsto \sigma_{Ju}\sigma_{v}(b)$ is an element of $\mathcal{B}^{A}(\mb{R}^{n}\times\mb{R}^{n})$ and we may define
\[ a\times_{J}b = \iint \sigma_{Ju}(a)\sigma_{v}(b) e^{2\pi i \, u\cdot v} \, du\, dv. \]

The homomorphism $A \longrightarrow C_{u}(\mb{R}^{n},A)$, $a\mapsto \widetilde{a}$, $\widetilde{a}(x) = \sigma_{x}(a)$, is equivariant for the respective actions $\sigma$ and $\tau$, thus maps $A^{\infty}\longrightarrow \mathcal{B}^{A}$. Moreover, $\widetilde{a\times_{J}b} = \widetilde{a}\times_{J}\widetilde{b}$, i.e. this is a homomorphism for the products $\times_{J}$. Thus $A^{\infty}$ is represented on $\mathcal{S}^{A}_{J}$, and we define a new norm $||\cdot||_{J}$ on $A^{\infty}$, $||a||_{J} = ||L_{\widetilde{a}}||$.

\begin{definition}[Rieffel deformation] Equip $A^{\infty}$ with the product $\times_{J}$ and the norm $||\cdot||_{J}$. This completion is denoted $A_{J}$ and is called the deformation of $A$ along $\sigma$ by $J$, or in short the \textit{Rieffel deformation} of $A$.
\end{definition}

Below we list some of the properties of the Rieffel deformation.
\renewcommand{\labelenumi}{(\roman{enumi})}
\begin{lemma}[Properties of the Rieffel deformation]\label{RieffelProperties} Let $A$ be a separable C*-algebra, $\sigma: \mb{R}^{n} \longrightarrow Aut(A)$ a strongly continuous action and $J\in M_{n}(\mathbb{R})$ such that $J^{t} = -J$.
\begin{enumerate}
  \item\label{RieffelProperties1} $\times_{J}$ is associative and the involution $*$ for $A$ is also an involution for $A_{J}$, which thus becomes a C*-algebra
  \item\label{RieffelProperties3} $a\times_{J}b = ab$ for $J=0$
  \item\label{RieffelProperties4} For every fixed point $a\in A^{\sigma}$, $a\times_{J}b = ab$ and $b\times_{J}a = ba$ for every $b\in A$
  \item\label{RieffelProperties5} $(A_{J})_{K} = A_{J+K}$ for any skew-symmetric $K\in Mat_{n}(\mathbb{R})$
  \item\label{RieffelProperties6} The action $\sigma$ is also an action on $A_{J}$, $\sigma: \mb{R}^{n}\longrightarrow Aut(A_{J})$. Moreover $(A_{J})^{\infty} = (A^{\infty})_{J}$
  \item\label{RieffelProperties7} The dense subalgebra $(A^{\infty})_{J}\subseteq A_{J}$ is stable under holomorphic functional calculus
  \item\label{RieffelProperties8} Given a $\sigma$-invariant ideal $I\subseteq A$, the equivariant short exact sequence
\[
\begin{CD}
0 @>>> I @>>> A @>>> A/I @>>> 0
\end{CD}
\]
implies a short exact sequence
\[
\begin{CD}
0 @>>> I_{J} @>>> A_{J} @>>> (A/I)_{J} @>>> 0
\end{CD}
\]
\item\label{RieffelProperties9} For any $T\in M_{n}(\mathbb{R})$, define a new action $\sigma^{T}$ by $\sigma^{T}_{x}(a) = \sigma_{Tx}(a)$, for $x\in\mb{R}^{n}$, $a\in A$. Performing the deformation procedure for the action $\sigma^{T}$ and skew-symmetric matrix $J$, denote by $\times_{J}^{T}$ the deformed product so obtained. Then
\[ \times^{T}_{J} = \times_{TJT^{t}}. \]
\end{enumerate}
\end{lemma}

The equivalence between the Rieffel deformation $A_{J}$ and the 2-cocycle deformation $A^{\psi_{J}}$ is given by the *-isomorphism of the following lemma. Recall that one considers the $\mb{R}^{n}$-product $(A\rtimes_{\sigma}\mb{R}^{n}, \lambda, \widehat{\sigma}^{\psi_{J}})$, the 2-cocycle $\psi_{J}$ in $(\ref{2cocycleEq})$ and $A^{\psi_{J}}\subseteq M(A\rtimes_{\sigma}\mb{R}^{n})$ is the subalgebra satisfying the Landstad conditions.

\begin{lemma}\label{RieffelKasprzakApproach} There is a *-isomorphism
\[ T: A^{\psi_{J}} \longrightarrow A_{J}. \]
\end{lemma}
\begin{proof} We refer the reader to \cite{H-M} for details, and give only the form of the *-isomorphism here.  Let $y\in C_{c}(\mb{R}^{n}, A^{\infty}) \subseteq A\rtimes_{\sigma}\mb{R}^{n}\subseteq M(A\rtimes_{\sigma}\mb{R}^{n})$, and suppose $y\in A^{\psi_{J}}$, which means that $\widehat{\sigma}^{\psi_{J}}_{x}(y) = y$ for all $x\in\mb{R}^{n}$. The isomorphism $T$ is described on such elements by
\[ T(y) = \int_{\mb{R}^{n}} y(v) \, dv. \]
\end{proof}

We consider $B = C([0,1])\otimes A$ with the action $\beta$ as in $(\ref{actionBeta})$. Note that $\beta_{x}(y)(s) = \sigma^{\sqrt{s}I}_{x}(y(s))$ (see Lemma $\ref{RieffelProperties} (ix)$).
For every $x\in\mb{R}^{n}$, let $\overline{\beta}_{x}\in Aut(M(B))$ denote the canonical extension of $\beta_{x}$ to the multiplier algebra, namely for $L\in M(B)$, $\overline{\beta}_{x}(L)(b) = \beta_{x}(L(\beta_{-x}(b)))$, for $b\in B$. A quick calculation reveals that for every $f\in C([0,1])$, $\overline{\beta}_{x}(\Phi_{B}(f)) = \Phi_{B}(f)$, i.e. $\Phi_{B}(C([0,1])) \subseteq M(B)^{\overline{\beta}}$. It is also clear that  $\Phi_{B}(C([0,1])) \subseteq M(B)^{\infty}$. From the inclusion $B\subseteq M(B)$ as a $\beta$-invariant ideal we get $B_{J}\subseteq M(B)_{J}$ by Lemma $\ref{RieffelProperties} \,\, (vii)$, and working inside $M(B)_{J}$ get from Lemma $\ref{RieffelProperties}\,\,(iii)$
\begin{equation}\label{deformFieldStructEq}
\Phi_{B}(f)\times_{J}y = \Phi_{B}(f)y = y\Phi_{B}(f) = y\times_{J}\Phi_{B}(f)
\end{equation}
for $y\in B^{\infty}$ and $f\in C([0,1])$, as $\Phi_{B}(f)\in M(B)^{\overline{\beta}}$. This yields a $C([0,1])$-algebra structure on $B_{J}$, denoted $\Phi_{B_{J}}: C([0,1])\longrightarrow ZM(B_{J})$ given by $\Phi_{B_{J}}(f)y = \Phi_{B}(f)\times_{J}y = \Phi_{B}(f)y$. As such, $B_{J}$ is an \textit{essential} $C([0,1])$-module, i.e. $\overline{C([0,1])B_{J}} = B_{J}$.

\begin{theorem} $(A_{tJ})_{t\in [0,1]}$ is a continuous field of C*-algebras, where we take as the algebra of sections $\Gamma( (A_{tJ})_{t\in [0,1]} )$ to be the algebra $B_{J}$.
\end{theorem}
\begin{proof} For each $s\in [0,1]$ let $K^{s} = I_{s}\otimes A$ be the ideal consisting of elements of $B = C([0,1],A)$ which vanish at the point $s$. Clearly, $B/K^{s} = A$. The short exact sequence
\[
\begin{CD}
0 @>>> K^{s} @>>> B @>>> A @>>> 0
\end{CD}
\]
is equivariant for $\beta$ acting on $K^{s}$ and $B$, and $\sigma^{\sqrt{s} 1}$ acting on $A$, so by Lemma $\ref{RieffelProperties}\,\,(vii)$ (cf. also Theorem 7.7 of \cite{Rieffel1}) we get a short exact sequence
\[
\begin{CD}
0 @>>> {K^{s}}_{J} @>>> B_{J} @>>> A^{\sqrt{s}1} _{J} @>>> 0
\end{CD}
\]

The fiber $(B_{J})_{s}$ over $s\in [0,1]$ of the $C([0,1])$-algebra $B_{J}$ is by definition the quotient $(B_{J})_{s} = B_{J}/(I_{s}B_{J})$. It is shown in \cite{Rieffel1} that ${K^{s}}_{J} = I_{s}B_{J}$, consequently $(B_{J})_{s} = B_{J}/(I_{s}B_{J}) = B_{J}/{K^{s}}_{J} = A^{\sqrt{s}1}_{J}$. Moreover, from Lemma $\ref{RieffelProperties}\,\,(viii)$ it follows that $A^{\sqrt{s}1}_{J} = A_{\sqrt{s}1J\sqrt{s}1} = A_{sJ}$, thus the bundle projection is $\pi_{s}: B_{J}\longrightarrow A_{sJ}$. Theorem 8.3 of \cite{Rieffel1} (see also Proposition 1.2 of \cite{Rieffel2}) establishes the continuity of the field $(A_{tJ})_{t\in [0,1]}$, for which $B_{J}$ is a maximal algebra of cross sections, henceforth denoted $\Gamma( (A_{tJ})_{t\in [0,1]} )$.
\end{proof}

Considering the *-isomorphism of Lemma $\ref{RieffelKasprzakApproach}$ at the level of bundles, we get
\begin{lemma}\label{Bundle-RieffelIsoKasprzak} The *-isomorphism
\[ T: B^{\psi} \longrightarrow B_{J} \]
is $C([0,1])$-equivariant, i.e. $T\circ\Phi_{B^{\psi}} = \Phi_{B_{J}}\circ T$.
\end{lemma}
\begin{proof} Let $b\in C_{c}(\mb{R}^{n}, B^{\infty}) \subset B\rtimes_{\beta}\mb{R}^{n} \subset M(B\rtimes_{\beta}\mb{R}^{n})$ be an element such that $\widehat{\beta}^{\psi}_{x}(b) = b$ for all $x\in\mb{R}^{n}$, i.e. $b$ is an element of $B^{\psi}$. The *-isomorphism is described on such elements by
\[ T(b) = \int_{\mb{R}^{n}}b(v). \]
Furthermore
\begin{align*}
T(\Phi_{B^{\psi}}(f)b) &= \int_{\mb{R}^{n}} (\Phi_{B^{\psi}}(f)b)(v) \,dv = \int_{\mb{R}^{n}} (\Phi_{B\rtimes_{\beta}\mb{R}^{n}}(f)b)(v)\, dv \\
&= \int_{\mb{R}^{n}} \Phi_{B}(f)(b(v))\, dv = \Phi_{B}(f) \int_{\mb{R}^{n}} b(v) \, dv,
\end{align*}
and since $\Phi_{B_{J}} = \Phi_{B}$ as in $(\ref{deformFieldStructEq})$, the claim follows.
\end{proof}

\begin{theorem}\label{maintheorem} Let $h\in [0,1]$. The evaluation map
\[ \pi_{h}: \Gamma( (A_{tJ})_{t\in [0,1]} ) \longrightarrow A_{hJ} \]
is a KK-equivalence.
\end{theorem}
\begin{proof} As $\Gamma( (A_{tJ})_{t\in [0,1]} ) = B_{J}$ is $C([0,1])$-linearly *-isomorphic to $B^{\psi}$, and $B^{\psi}$ is an RKK-fibration (Theorem $\ref{theoremRKKfibration}$), thus $\Gamma( (A_{tJ})_{t\in [0,1]} )$ is an $RKK$-fibration and hence a $KK$-fibration (Lemma $\ref{RKKfibKK}$). So for any $f: \Delta^{p}\longrightarrow [0,1]$ and $v\in\Delta^{p}$, the quotient map $q_{v}: f^{*}(\Gamma( (A_{tJ})_{t} )) \longrightarrow A_{f(v)J}$ is a $KK$-equivalence. We take the identity function of the $1$-simplex, namely $f: \Delta^{1}=[0,1]\longrightarrow [0,1]$, $f(s) = s$. Then $f^{*}(\Gamma( (A_{tJ})_{t} )) = \Gamma( (A_{tJ})_{t} )$, $q_{h} = \pi_{h}$ and
\[ \pi_{h}: \Gamma( (A_{tJ})_{t} ) \longrightarrow A_{hJ } \]
is a $KK$-equivalence, for every $h\in [0,1]$.
\end{proof}

\section{Comments}
\subsection{Theta deformation}
Here we discuss a special case of Rieffel deformation, namely \textit{theta deformation} and one possible variation to the above approach to KK-equivalence by bundle methods. Theta deformation concerns a separable C*-algebra $A$ on which there is a strongly continuous action of the $n$-torus, $\sigma: \mathbb{T}^{n}\longrightarrow Aut(A)$, with a given skew-symmetric matrix $\theta \in M_{n}(\mathbb{R})$. This is just a special case of Rieffel deformation in which the $n$-torus is regarded as the quotient $\mathbb{T}^{n} = \mb{R}^{n}/2\pi\mb{Z}^{n}$, and one obtains the deformed algebra $A_{\theta}$. An alternative and perhaps more direct picture can be given by following \cite{Connes-DuboisViolette}. First define $C(\mathbb{T}^{n}_{\theta})$ to be the unital C*-algebra generated by  unitaries $u_{1},\ldots, u_{n}$ with relations
\[ u_{j}u_{k} = e^{2\pi i \theta_{j,k}}u_{k}u_{j}, \mbox{ for } j, k = 1,\ldots, n. \]
(Note that this is just the Rieffel deformation $C(\mathbb{T}^{n})_{\theta}$ of the commutative C*-algebra $C(\mathbb{T}^{n})$ with respect to the translation action of the $n$-torus; the notation $C(\mathbb{T}^{n}_{\theta})$ is suggestive of the terminology of "noncommutative manifolds" as in \cite{Connes-DuboisViolette}). On $C(\mathbb{T}^{n}_{\theta})$ there is the action $\tau: \mathbb{T}^{n} \longrightarrow Aut(C(\mathbb{T}^{n}_{\theta}))$, $\tau_{s}(u_{j}) = e^{2\pi i s_{j}}u_{j}$, for $s\in\mathbb{T}^{n}$. By considering the diagonal action $\sigma\otimes\tau^{-1}: \mathbb{T}^{n}\longrightarrow Aut(A\otimes C(\mathbb{T}^{n}_{\theta}))$ one defines the theta deformed algebra
\begin{equation}\label{thetaDeformation}
A_{\theta} = (A\otimes C(\mathbb{T}^{n}_{\theta}))^{\sigma\otimes\tau^{-1}}
\end{equation}
as the fixed-point C*-subalgebra for this diagonal action.

We shall define a continuous C*-bundle over $[0,1]$ whose fiber over $t\in [0,1]$ will not be $A_{t\theta}$ per se, but will be strongly Morita equivalent to it. The benefit of this particular bundle will be that the evaluation map will easily be seen to yield a KK-equivalence element, and the remaining KK-equivalence is then given by the strong Morita equivalence. First we record the result we need regarding the strong Morita equivalence.

\begin{lemma} $A_{\theta} \sim_{M} A\rtimes_{\sigma}\mathbb{T}^{n}\rtimes_{\gamma_{1}}\mathbb{Z}\rtimes\cdots\rtimes_{\gamma_{n}}\mathbb{Z}$.
\end{lemma}
\begin{proof} By results of $\cite{Ng}$ we get the strong Morita equivalence
\[ (A\otimes C(\mathbb{T}^{n}_{\theta}))^{\sigma\otimes\tau^{-1}} \sim_{M} (A\otimes C(\mathbb{T}^{n}_{\theta}))\rtimes_{\sigma\otimes\tau^{-1}}\mathbb{T}^{n}. \]
The latter crossed product algebra is *-isomorphic to the crossed product in the statement of the lemma, which we now define. Let $\gamma_{1} \in Aut(A\rtimes_{\sigma}\mathbb{T}^{n})$ be $\gamma_{1}(g)(s) = e^{2\pi i s_{1}}g(s)$ for $g\in A\rtimes_{\sigma}\mathbb{T}^{n}$ and let $u_{1}$ be the implementing unitary. Proceed inductively to define actions $\gamma_{2},\ldots,\gamma_{n}$ with implementing unitaries $u_{2},\ldots,u_{n}$ such that
\begin{equation}\label{GammaActions}
\gamma_{j}(g)(s) = e^{2\pi i s_{j}}g(s), \quad \gamma_{j}(u_{k}) = e^{2\pi i \theta_{j,k}}u_{k}, \, j<k,
\end{equation}
so the covariance relation $\gamma_{j}(u_{k}) = u_{j}u_{k}u_{j}^{*} = e^{2\pi i \theta_{j,k}}u_{k}$ means precisely $u_{j}u_{k} = e^{2\pi i \theta_{j,k}}u_{k}u_{j}$. The *-isomorphism $ (A\otimes C(\mathbb{T}^{n}_{\theta}))\rtimes_{\sigma\otimes\tau^{-1}}\mathbb{T}^{n} \longrightarrow A\rtimes_{\sigma}\mathbb{T}^{n}\rtimes_{\gamma_{1}}\mathbb{Z}\rtimes\cdots\rtimes_{\gamma_{n}}\mathbb{Z}$ can be explicitly described on the dense *-subalgebra $A\otimes C(\mathbb{T}^{n}_{\theta})\otimes C(\mathbb{T}^{n})$ as $a\otimes u_{j}\otimes h \mapsto u_{j}(ah)$ where one understands $ah\in A\otimes C(\mathbb{T}^{n})\subseteq A\rtimes_{\sigma}\mathbb{T}^{n}$.
\end{proof}

Let $B = C([0,1])\otimes A\rtimes_{\sigma}\mathbb{T}^{n} = C([0,1], A\rtimes_{\sigma}\mathbb{T}^{n})$. We may decompose $\sigma$ into its coordinate actions $\sigma_{1},\ldots,\sigma_{n}$ where $\sigma_{j}(z) = \sigma_{(1,\ldots,z,\ldots,1)}$ for $z\in\mathbb{T}$. For $j,k=1,\ldots,n$ let $h_{j,k}\in C([0,1])$ be the function
\[ h_{j,k}(t) = e^{2\pi i t\theta_{j,k}}. \]
Define $\alpha_{1}\in Aut(B)$ by
\[ \alpha_{1}(f\otimes g) = f\otimes \widehat{\sigma_{1}}(g), \quad f\in C([0,1]), g\in A\rtimes_{\sigma}\mathbb{T}^{n} \]
and let $v_{1}$ be the unitary implementing $\alpha_{1}$ in $B\rtimes_{\alpha_{1}}\mathbb{Z}$. Define $\alpha_{2}\in Aut(B\rtimes_{\alpha_{1}}\mathbb{Z})$ by
\[ \alpha_{2}( (f\otimes g)v_{1}^{m} ) = (h_{1,2}^{m}f\otimes \widehat{\sigma_{2}}^{m}(g))v_{1}^{m}, \quad m\in\mathbb{Z}. \]
Proceeding inductively we thus obtain actions $\alpha_{1},\ldots,\alpha_{n}$ with respective implementing unitaries $v_{1},\ldots,v_{n}$,
\[ \alpha_{k}( (f\otimes g)v_{j}^{m} ) = v_{k}( (f\otimes g)v_{j}^{m} )v_{k}^{*} = (h_{j,k}^{m}f\otimes\widehat{\sigma_{k}}^{m}(g))v_{j}^{m}. \]
Let $\pi_{t}: C([0,1])\otimes A\rtimes_{\sigma}\mathbb{T}^{n} \longrightarrow A\rtimes_{\sigma}\mathbb{T}^{n}$ be the evaluation map, $\pi_{t}(f\otimes g) = f(t)g$.
For each $t\in [0,1]$, starting with $A\rtimes_{\sigma}\mathbb{T}^{n}$ inductively define actions $\gamma_{1}^{t},\ldots,\gamma_{n}^{t}$ as in $(\ref{GammaActions})$ with respective unitaries $u_{1},\ldots,u_{n}$ such that
\[ \gamma_{j}^{t}(g)(s) = e^{2\pi i s_{j}}g(s), \quad \gamma_{j}^{t}(u_{k}) = e^{2\pi i t\theta_{j,k}}u_{k}. \]
Note that the actions $\gamma_{j}$ of $(\ref{GammaActions})$ are just $\gamma_{j} = \gamma_{j}^{1}$ with $t=1$. Furthermore, $\pi_{t}\circ\alpha_{1} = \gamma_{1}^{t}\circ\pi_{t}$, i.e. $\pi_{t}$ is a $\mathbb{Z}$-equivariant *-homomorphism between the C*-dynamical systems and so passes to a *-homomorphism between the crossed products
\begin{equation}\label{sec5eqBundleStep1}
\pi_{t}: (C([0,1])\otimes A\rtimes_{\sigma}\mathbb{T}^{n})\rtimes_{\alpha_{1}}\mathbb{Z} \longrightarrow A\rtimes_{\sigma}\mathbb{T}^{n}\rtimes_{\gamma_{1}^{t}}\mathbb{Z},
\end{equation}
which is a continuous C*-bundle. Iterating this, one has $\pi_{t}\circ\alpha_{j} = \gamma_{j}^{t}\circ\pi_{t}$ for each $j=1,\ldots,n$, where $\pi_{t}$ is understood on the appropriate crossed product. Thus we get a continuous C*-bundle
\begin{equation}\label{sec5eqBundle}
\pi_{t}: C([0,1], A\rtimes_{\sigma}\mathbb{T}^{n})\rtimes_{\alpha_{1}}\mathbb{Z}\rtimes\cdots\rtimes_{\alpha_{n}}\mathbb{Z} \longrightarrow A\rtimes_{\sigma}\mathbb{T}^{n}\rtimes_{\gamma_{1}^{t}}\mathbb{Z}\rtimes\cdots\rtimes_{\gamma_{n}^{t}}\mathbb{Z}.
\end{equation}

For each $t\in [0,1]$ let $I_{t} = \{f\in C([0,1]) \, \lvert \quad f(t) = 0\}$ be the ideal of functions vanishing at the point $t$. The ideal $I_{t}\otimes A\rtimes_{\sigma}\mathbb{T}^{n} \subseteq C([0,1])\otimes A\rtimes_{\sigma}\mathbb{T}^{n}$ is $\alpha_{1}$-invariant, so it follows that the kernel of the *-homomorphism $\pi_{t}$ in $(\ref{sec5eqBundleStep1})$ is
\[ ker\,\pi_{t} = (I_{t}\otimes A\rtimes_{\sigma}\mathbb{T}^{n})\rtimes_{\alpha_{1}}\mathbb{Z}. \]
By iteration, it follows that the kernel of the *-homomorphism $\pi_{t}$ in $(\ref{sec5eqBundle})$ is
\[ ker\,\pi_{t} = (I_{t}\otimes A\rtimes_{\sigma}\mathbb{T}^{n})\rtimes_{\alpha_{1}}\mathbb{Z}\rtimes\cdots\rtimes_{\alpha_{n}}\mathbb{Z}. \]

Using a homeomorphism of $[0,1]$ to itself, mapping $t$ to $1$, there is a *-isomorphism $I_{t} \cong C_{0}([0,1))$. This means $I_{t}\otimes A\rtimes_{\sigma}\mathbb{T}^{n} \cong C_{0}([0,1))\otimes A\rtimes_{\sigma}\mathbb{T}^{n} = Cone(A\rtimes_{\sigma}\mathbb{T}^{n})$, hence
\begin{equation}\label{kerCone}
ker\,\pi_{t} = Cone(A\rtimes_{\sigma}\mathbb{T}^{n})\rtimes_{\alpha_{1}}\mathbb{Z}\rtimes\cdots\rtimes_{\alpha_{n}}\mathbb{Z}.
\end{equation}

We recall a few general facts which we will appeal to shortly, in particular contractibility of cones and the Pimsner-Voiculescu six-term exact sequence. First, a C*-algebra $B$ is called \textit{KK-contractible} if $KK(B,B) = 0$. This also implies $KK(B,D) = 0 = KK(D,B)$ for any other C*-algebra $D$.

Suppose there is an action $\beta\in Aut(B)$. The Pimsner-Voiculescu six-term exact sequence in KK-theory is
\[
\begin{CD}
KK(D,B) @>1-\beta_{*}>> KK(D,B) @>>> KK(D,B\rtimes_{\beta}\mathbb{Z}) \\
@AAA @. @VVV \\
KK^{1}(D, B\rtimes_{\beta}\mathbb{Z})  @<<< KK^{1}(D,B) @<<1-\beta_{*}< KK^{1}(D,B)
\end{CD}
\]
Observe that if $B$ is KK-contractible, then the six-term exact sequence reads
\[
\begin{CD}
0 @>1-\beta_{*}>> 0 @>>> KK(D,B\rtimes_{\beta}\mathbb{Z}) \\
@AAA @. @VVV \\
KK^{1}(D, B\rtimes_{\beta}\mathbb{Z})  @<<< 0 @<<1-\beta_{*}< 0
\end{CD}
\]
and using in particular $D = B\rtimes_{\beta}\mathbb{Z}$ we deduce $KK(B\rtimes_{\beta}\mathbb{Z}, B\rtimes_{\beta}\mathbb{Z}) = 0$, i.e. $B\rtimes_{\beta}\mathbb{Z}$ is KK-contractible.

Given any separable C*-algebra $D$, its cone $Cone(D) = C_{0}([0,1))\otimes D$ is KK-contractible.

\begin{theorem} For every $t\in [0,1]$ the bundle map
\[ \pi_{t}: C([0,1], A\rtimes_{\sigma}\mathbb{T}^{n})\rtimes_{\alpha_{1}}\mathbb{Z}\rtimes\cdots\rtimes_{\alpha_{n}}\mathbb{Z} \longrightarrow A\rtimes_{\sigma}\mathbb{T}^{n}\rtimes_{\gamma_{1}^{t}}\mathbb{Z}\rtimes\cdots\rtimes_{\gamma_{n}^{t}}\mathbb{Z} \]
gives a KK-equivalence.
\end{theorem}
\begin{proof} From $(\ref{kerCone})$ $ker\,\pi_{t} = Cone(A\rtimes_{\sigma}\mathbb{T}^{n})\rtimes_{\alpha_{1}}\mathbb{Z}\rtimes\cdots\rtimes_{\alpha_{n}}\mathbb{Z}$. Then the KK-contractibility of $Cone(A\rtimes_{\sigma}\mathbb{T}^{n})$ combined with a repeated Pimsner-Voiculescu six-term sequence argument as above establishes that $ker\,\pi_{t}$ is KK-contractible. This implies that $\pi_{t}$ gives a KK-equivalence element.
\end{proof}

\subsection{Invariance of the index}
The index pairing is the pairing between K-theory and K-homology
\[ K_{0}(A)\times K^{0}(A) \longrightarrow \mathbb{Z} \]
\begin{equation}\label{indexParingEq}
\langle [e], [(\mathcal{H},F)]\rangle = index\left(e(F^{+}\otimes 1_{k})e: e\mathcal{H}^{k} \longrightarrow e\mathcal{H}^{k}\right),
\end{equation}
for a projection $e\in M_{k}(A)$ and Fredholm module $(\mathcal{H},F)$ for $A$. This pairing is nothing but the KK-product
\begin{equation}\label{indexPairingKKprod}
KK(\mathbb{C},A) \times KK(A,\mathbb{C}) \longrightarrow KK(\mathbb{C},\mathbb{C})
\end{equation}
after the identifications $K_{0}(A) = KK(\mathbb{C},A)$, $K^{0}(A) = KK(A,\mathbb{C})$ and $KK(\mathbb{C},\mathbb{C}) = \mathbb{Z}$.

See also \cite{Yamashita} for a discussion of theta deformation and the invariance of the index, and moreover a calculation of the Chern character map for the deformation.

Given an even spectral triple $(A,\mathcal{H},D)$ there is the associated Fredholm module $(\mathcal{H}, F)$ with $F = D|D|^{-1}$.

Our separable C*-algebra $A$ is assumed equipped with an action $\sigma:  \mathbb{T}^{n}\longrightarrow Aut(A)$, and let $\mathcal{A}\subseteq A$ be the dense *-subalgebra of smooth elements for the action. Suppose $(\mathcal{A},\mathcal{H},D)$ is a spectral triple, with a *-representation $\varphi: A\longrightarrow B(\mathcal{H})$. Assume the action to be unitarily implemented by $U:\mathbb{T}^{n} \longrightarrow B(\mathcal{H})$, $\varphi(\sigma_{s}(a)) = U_{s}\varphi(a)U_{s}^{*}$, and that $U_{s}D = DU_{s}$ for each $s\in\mathbb{T}^{n}$.

Theta deformation is an \textit{isospectral} deformation, meaning that the same data $(\mathcal{H}, D)$ which describes a noncommutative geometry for $A$, is also taken to serve a noncommutative geometry for $A_{\theta}$. In order to study these aspects, it is useful to work with the following picture of the deformation. Any element $a\in \mathcal{A}$ decomposes into a norm convergent series $a = \sum_{r\in\mb{Z}^{n}} a_{r}$ where each $a_{r}\in \mathcal{A}$ satisfies $\sigma_{s}(a_{r}) = e^{-2\pi i r\cdot s}a_{r}$, for $s\in\mathbb{T}^{n}$. Given two elements $a, b\in \mathcal{A}$ with decompositions $a = \sum_{r}a_{r}$ and $b=\sum_{p}b_{p}$, the product $\times_{\theta}$ takes the form
\begin{equation}\label{thetaProd}
a_{r}\times_{\theta}b_{p} = e^{2\pi i r\cdot\theta p}a_{r}b_{p}
\end{equation}
between two component elements $a_{r}$ and $b_{p}$. The product $a\times_{\theta}b$ is then the linear extension of the componentwise product $(\ref{thetaProd})$. The *-algebra $\mathcal{A}_{\theta}$ is just $\mathcal{A}$ equipped with this product. The correspondence with the definition in $(\ref{thetaDeformation})$ is just
\[ a_{r} \mapsto a_{r} \otimes u_{1}^{r_{1}}\cdots u_{n}^{r_{n}} \in (A\otimes C(\mathbb{T}^{n}_{\theta}))^{\sigma\otimes\tau^{-1}}. \]
We have a representation $\varphi_{\theta}$ of $\mathcal{A}_{\theta}$ on the same Hilbert space, $\varphi_{\theta}: \mathcal{A}_{\theta} \longrightarrow B(\mathcal{H})$, by
\[ \varphi_{\theta}(a) = \sum_{r} \varphi(a_{r})U_{q(\theta r)}, \]
$A_{\theta}$ is then the norm closure and $(\mathcal{A}_{\theta}, \mathcal{H}, D_{\theta})$ is the deformed spectral triple, where $D_{\theta} = D$.

For the even spectral triple $(A,\mathcal{H},D)$ we shall denote by $[D] = [(\mathcal{H}, \varphi, F)]\in K^{0}(A)$ the corresponding element of K-homology. Likewise we denote by $[D_{\theta}] = [(\mathcal{H}, \varphi_{\theta}, F)]\in K^{0}(A_{\theta})$ the element associated to the spectral triple $(A_{\theta},\mathcal{H},D_{\theta})$.

\begin{corollary} The KK-equivalence of Theorem $\ref{maintheorem}$ induces an isomorphism $K^{0}(A) \cong K^{0}(A_{\theta})$ mapping
 $[D] \mapsto [D_{\theta}]$.
\end{corollary}
\begin{proof} Let $\Gamma = \Gamma( (A_{t\theta})_{t\in[0,1]} )$. From the bundle maps $\pi_{0}: \Gamma\longrightarrow A$ and $\pi_{1}: \Gamma\longrightarrow A_{\theta}$ we get by Theorem $\ref{maintheorem}$ the KK-equivalence elements $[\pi_{0}]\in KK(\Gamma, A)$ and $[\pi_{1}]\in KK(\Gamma, A_{\theta})$. The relevant mappings between KK-groups is described by the KK-products
\[
\begindc{\commdiag}
\obj(3,3)[L1]{$KK(\Gamma,\mathbb{C})$}
\obj(1,1)[L2]{$KK(A,\mathbb{C})$}
\obj(5,1)[L3]{$KK(A_{\theta},\mathbb{C})$}
\mor{L2}{L1}{\scriptsize$[\pi_{0}]\cdot$}
\mor{L3}{L1}{\scriptsize$[\pi_{1}]\cdot$}[-1,0]
\mor{L2}{L3}{\scriptsize$[\pi_{1}]^{-1}\cdot [\pi_{0}]\cdot$}[-1,1]
\enddc
\]
where $[\pi_{0}] = [(A, \pi_{0}, 0)] \in KK(A,\mathbb{C})$ and $[\pi_{1}] = [(A_{\theta}, \pi_{1}, 0)]\in KK(\Gamma,A_{\theta})$ are the KK-cycle descriptions.

The element $[D] = [(\mathcal{H},\varphi, F)]\in KK(A,\mathbb{C})$ is the element canonically associated to the given spectral triple $(A,\mathcal{H},D)$ as explained above, and upon taking the KK-product we get
\begin{equation}\label{indexEq1}
[\pi_{0}]\cdot [D] = [(\mathcal{H},\varphi\circ\pi_{0}, F)]\in KK(\Gamma, \mathbb{C}).
\end{equation}
Likewise $[D_{\theta}] = [(\mathcal{H}, \varphi_{\theta}, F)]\in KK(A_{\theta},\mathbb{C})$ is the element associated to the deformed spectral triple $(A_{\theta},\mathcal{H},D_{\theta})$, and the KK-product is then
\begin{equation}\label{indexEq2}
[\pi_{1}]\cdot [D_{\theta}] = [(\mathcal{H},\varphi\circ\pi_{1}, F)]\in KK(\Gamma, \mathbb{C}).
\end{equation}
It will be enough to establish the equality $[\pi_{0}]\cdot [D] = [\pi_{1}]\cdot [D_{\theta}]$ in $KK(\Gamma,\mathbb{C})$. This follows from homotopy of KK-cycles.
Indeed, let $(E,\phi, F)\in KK(\Gamma, I\mathbb{C})$ be the element where $E = C([0,1],\mathcal{H})$, $\phi: \Gamma\longrightarrow \mathcal{L}_{I\mathbb{C}}(E)$, $(\phi(s)\xi)(t) = s(t)\xi(t)$, and $I\mathbb{C} = C([0,1])\otimes\mathbb{C} = C([0,1])$. Let $ev_{0}$ and $ev_{1}$ denote the respective evaluation morphisms $E\longrightarrow \mathcal{H}$. It is easy to check (using details explained in \cite{Connes-Landi}) that $(E,\phi,F)$ provides a homotopy between the KK-cycles $(\ref{indexEq1})$ and $(\ref{indexEq2})$, i.e. isomorphisms of the KK-cycles with the pushouts of $ev_{0}$ and $ev_{1}$ respectively,
\[ (E_{ev_{0}}, \phi_{ev_{0}}, F_{ev_{0}}) \cong [(\mathcal{H},\varphi\circ\pi_{0}, F)] \quad\mbox{ and }\quad (E_{ev_{1}}, \phi_{ev_{1}}, F_{ev_{1}}) \cong [(\mathcal{H},\varphi\circ\pi_{1}, F)]. \]
\end{proof}

The KK-equivalence of Theorem $\ref{maintheorem}$ implies the isomorphisms
\[ K_{0}(A) = KK(\mathbb{C}, A) \longrightarrow KK(\mathbb{C},A_{\theta}) = K_{0}(A_{\theta}), \quad [e] \longmapsto [e]\cdot [\pi_{0}]^{-1}\cdot [\pi_{1}], \]
and
\[ K^{0}(A) = KK(A, \mathbb{C}) \longrightarrow KK(A_{\theta},\mathbb{C}) = K^{0}(A_{\theta}), \quad [(\mathcal{H},F)] \longmapsto [\pi_{1}]^{-1}\cdot [\pi_{0}]\cdot [(\mathcal{H},F)], \]
and regarding the index pairing $(\ref{indexParingEq})$ or equivalently the KK-product $(\ref{indexPairingKKprod})$, we get
\[
\begin{CD}
K_{0}(A)\times K^{0}(A) @>\mbox{index}>> \mathbb{Z} \\
@VVV @VV{\lvert\lvert}V \\
K_{0}(A_{\theta})\times K^{0}(A_{\theta}) @>>\mbox{index}> \mathbb{Z}
\end{CD}
\]
where $[e]\cdot [(\mathcal{H},F)]$ is the top index pairing and
\[ [e]\cdot [\pi_{0}]^{-1}\cdot [\pi_{1}] \cdot [\pi_{1}]^{-1}\cdot [\pi_{0}]\cdot [(\mathcal{H},F)] = [e]\cdot [(\mathcal{H},F)] \]
is the bottom index pairing after having followed the isomorphisms induced by the KK-equivalences.

\newpage

redpajama_set_name: RedPajamaArXiv


text:
\section{ Introduction}
The spectra of non-Hermitian systems lie in the 2D complex plane, and can exhibit intriguing geometric and topological spectral transitions~\cite{berry2004EP,PhysRevLett.80.5243,experimentzeuner2015,experimentHodaei2017,experimentChen2017,experimentzhu2018,experimentgal2018,experimentmiguel2018,experimentli2019,experimentwu2019,lee2020exceptional,li2021quantized,circuitPhysRevLett.126.215302,PhysRevX.8.031079,IPRPhysRevA.95.022117,RevModPhys.93.015005}.
In particular, it is known~ \cite{PhysRevLett.124.086801,PhysRevLett.124.056802,PhysRevLett.125.126402,PhysRevB.104.125109}
 that if the spectrum under periodic boundary conditions (PBCs) is a loop that encloses a nonvanishing region, the spectrum of the same system under semi-infinite boundary conditions (SIBCs) will fill up the interior of this loop.
 This intriguing fact is due to the non-local nature of the non-Hermitian skin effect (NHSE), which has inspired numerous theoretical and experimental~\cite{circuitliu2021non,experimentsebastian2020,xiao2020non, experimentlin2021steering,circuithofmann2020reciprocal,experimenthelbig2020,circuitPhysRevLett.122.247702,experimentBudich2020,PhysRevResearch.2.013058,circuithofmann2020reciprocal,PhysRevApplied.14.064076,PhysRevResearch.2.013280,circuitrafi2021non,experimentMcDonald2020,circuithofmann2020reciprocal,circuitzou2021observation,zhang2021observation} developments and challenged various longheld paradigms in physics.
The NHSE, which arises in non-Hermitian lattices with broken reciprocity, amplifies and pumps all states towards a boundary, such that the effects of boundary hoppings become non-perturbatively large~\cite{PhysRevLett.121.086803,PhysRevResearch.1.023013,PhysRevLett.124.066602,PhysRevLett.125.118001,PhysRevB.102.085151,PhysRevLett.121.086803,PhysRevResearch.1.023013,Longhi:20,li2020critical,PhysRevLett.125.118001,PhysRevLett.124.066602,PhysRevB.104.L161106,PhysRevB.99.201103,circuitMotohiko20191,PhysRevLett.121.086803,PhysRevB.102.205118,PhysRevLett.125.118001,PhysRevB.102.241202,PhysRevLett.124.250402,PhysRevB.102.201103,PhysRevB.104.L161106,lee2021many,PhysRevB.103.075126,PhysRevB.104.125416,PhysRevLett.126.176601,li2021impurity,PhysRevLett.127.116801,circuitxuke2021,PhysRevB.104.L161106,PhysRevB.103.085428,PhysRevB.104.165117,zhang2021universal,yoshida2019mirror,PhysRevLett.124.250402,Lee2019hybrid,zhang2020non,PhysRevB.102.085151,luo2020skin,YUCE2020126094,PhysRevB.99.245116,PhysRevB.100.075403,luo2020skin,PhysRevLett.125.186802}.

Intuitively, a semi-infinite 1D lattice system can have a spectrum that fills up a 2D region because its eigenstates only need to satisfy boundaries conditions on one side, and are hence free to accumulate against it with any spatial decay length. As such, an eigenstate is characterized by two continuous variables: its wavenumber and decay length, the latter which possess no Hermitian analog.
 However, true semi-infinite systems can neither be numerically nor experimentally simulated, and their non-Hermitian properties have so far been mathematical curiosities.

In this work, we show how to construct finite 1D systems whose spectra nevertheless fills up the 2D interiors of their PBC spectra. This is achieved with appropriately designed disordered couplings which mathematically simulate the effects of semi-infinite boundaries, namely the co-existence of a continuum of different decay length scales of the skin eigenstates. Notably, the density of states in the 2D complex plane can be fine-tuned towards a variety of desired profiles by tuning the disorder distribution. While it is arguably easy to fill up a 2D spectral region with the eigenenergies of many separate (uncoupled) 1D chains, doing so with a \emph{single} chain is nontrivial due to the non-local effects of NHSE accumulation that can propagate across very distant parts of the chain~\cite{li2021impurity,PhysRevLett.127.116801}. As such, our construction can be construed as a stochastic means to subtly control the distribution of skin decay lengths, and also the propagation of skin accumulation tendencies.

\section{Exploring the complex energy plane by tuning boundary conditions}

The starting point of our work is the observation that, by modifying the boundary conditions of a non-Hermitian system with unbalanced hoppings, we can access a continuum of complex energy spectra, and thus sweep across the interior of the PBC loop~\cite{PhysRevLett.126.176601,PhysRevB.103.L140201,li2021impurity,PhysRevLett.127.116801}. This is thanks to the non-Hermitian lattice's extreme sensitivity to the boundary conditions, a phenomenon commonly known as the non-Hermitian skin effect. In the extreme limit of open boundary conditions (OBCs), the eigenenergy continuum are given by
\begin{equation}\label{eq1-1}
\begin{split}
\bar E(k)=E(k+i\kappa_{\text{c}})\ ,
\end{split}
\end{equation}
where $E(k)$ is the dispersion of the original Hamiltonian $H(k)$, and $\kappa_{\text{c}}$ is the imaginary part of the momentum that represents the boundary localization of former bulk eigenstates~ \cite{PhysRevB.104.125109,circuitliu2021non,experimentsebastian2020,xiao2020non,PhysRevResearch.1.023013,PhysRevLett.124.066602,PhysRevLett.125.118001,PhysRevB.102.085151,PhysRevLett.123.170401,PhysRevLett.123.066404}
, which now decay like $\sim e^{-\kappa_\text{c}x}$. In general, $\kappa_{\text{c}}$ can be a complicated function of $k$, and is determined by the condition that $\bar E(k)$, $k\in [0,2\pi)$ does not enclose any nonzero area~\cite{grein1971multiconfiguration,PhysRevLett.121.086803,PhysRevLett.123.066404,PhysRevLett.123.246801,PhysRevResearch.1.023013,xiao2020non,PhysRevLett.124.066602,PhysRevLett.125.226402,wu2021connections,guo2021non,experimentlin2021steering}. But for the purpose of this work, $\kappa_\text{c}$ remains as a constant.

  Interpolating $\kappa$ from $\kappa_{\text{c}}$ to $0$, the energy spectrum $E(k+i\kappa)$ will start off as the OBC energy spectrum, and continuous evolve till it coincides with the PBC spectrum at $\kappa=0$, in the process passing through the entire region enclosed by the PBC loop. Below, we describe how to physically access such intermediate values of $\kappa$ by tuning the boundary hoppings.

  Using the Hatano-Nelson model~\cite{PhysRevX.8.031079,PhysRevB.104.L161106,NHMPhysRevLett.77.570,NHMhatano1997vortex,PhysRevB.92.094204,PhysRevB.103.L140201} for convenient illustration, since it only has nearest neighbor unbalancing hoppings, we have
  \begin{equation}\label{eq1-2}
\begin{split}
H=\sum^{L-1}_{n=1} t |n\rangle\langle n+1|+   t'|n+1\rangle\langle n|& \\
+\mu |L \rangle\langle 1|+ \mu' |1\rangle\langle L|&\  ,
\end{split}
\end{equation}
 where $t/ t'$ is the left/right hopping amplitude and $\mu,\mu'$ represent the boundary hoppings that connect the left and right boundaries.
Writing an eigenstate as $|\psi\rangle=\sum_n\psi_n|n\rangle$ with $H|\psi\rangle=E|\psi\rangle$, the wave function amplitude is given by $\psi_n $ at the $n$-th site in the lattice.
For a lattice comprising $L$ sites ($n = 1, 2, 3, ..., L $), the matrix Hamiltonian $H$ in Eq.\eqref{eq1-2} gives rise to bulk and boundary conditions which together give (See Appendix \ref{apppendix1}):
\begin{equation}\label{eq1-3}
\begin{split}
&\left(tt'\left(\frac{t'}{t}\right)^{L}\frac{1}{z^L}- t' \mu\right)\left(t\mu' z^{L}-tt'\right) \\
&\ =z^2\left(t\mu' \left(\frac{t'}{t}\right)^{L}\frac{1}{z^L}-tt'  \right)\left(t^2z^L-t \mu \right)\ ,
\end{split}
\end{equation}
with $E=tz+t'/z$ and $z=\exp(-\kappa+ik)$. Solving this characteristic equation gives $\kappa$, as shown in TABLE.\ref{tab1}, which can then be substituted into $\bar E=E(k+i\kappa)$ to yield the spectrum (See Appendix \ref{apppendix1}). It is seen that under OBCs, $\kappa$ only depends on the relative amplitudes $t/t'$ of the left/right hoppings. However, when the boundary and bulk hoppings are not equal, $\kappa$ will depend either on $t/\mu$ or $t'/\mu'$, depending on the relative strength of $t$ and $t'$ as well as whether $\mu\mu'$ or $tt'$ is larger.

\begin{table}[h!]
\renewcommand\arraystretch{1.5}
  \centering
  \begin{tabular}{c|c|c}
\hline
\hline
 &$t'<t$& $t'>t$\\
 \hline
\multirow{2}{*}{$\mu\mu' < tt'$}  &$ \kappa=\log\left({t/\mu}\right)/L$
&$ \kappa=\log\left({\mu'/t'}\right)/L$\\
&$(\kappa\in[0,\kappa_{\text{c}}])$
&$(\kappa\in[\kappa_{\text{c}},0])$\\
\hline
$\mu\mu' > tt'$ &$ \kappa=\log\left({\mu'/t'}\right)/L$&$\kappa=\log\left({{t/\mu}}\right)/L$\\
\hline
 \multicolumn{3}{l}{PBC \ ($\mu=t,\mu'=t'$): \qquad\  $\kappa=0$}\\
 \multicolumn{3}{l}{OBC\ ($\mu=\mu'=0$):\qquad\  $\kappa=\kappa_c=1/2\log(t/t')$,}\\
\hline
\hline
\end{tabular}
\caption{  \label{tab1} Approximate analytic forms of $\kappa$ of the model Eq.\eqref{eq1-1} in various regimes, for large $L$. Here, $\kappa_\text{c}=1/2\log(t/t')$, the value of $\kappa$ where the spectrum collapses into the OBC spectrum that encloses zero area. With appropriate tuning of boundary hoppings $\mu,\mu'$, we will be able to obtain any $\kappa$ that lies between the PBC and OBC cases i.e. $[0,\kappa_\text{c}]$.\\
}
\end{table}

 The upshot is that if we take the boundary hoppings $\mu,\mu'$ to be random numbers from $(0,t)$ and $(0,t')$,
 the energy spectrum will fall within the PBC spectral loop and, after multiple random trials, the combined energy spectra will fill up the loop, as shown in Fig.\ref{fig:1}(b). So far, this is not very surprising, since we are using an ensemble of 1D systems to fill up a 2D region. In the following sessions, we shall demonstrate how we can instead construct a \emph{single} 1D system whose eigenenergies fill up the interior of a spectral loop.
 \begin{figure}[htp]
    \begin{centering}
    \includegraphics[width=1\linewidth]{figure1.pdf}
    \par\end{centering}
    \protect\caption{\label{fig:1}
\textbf{Filling up a spectral ellipse with a large number of separate Hatano-Nelson chains with random boundary hoppings.} (a1-a2) shows the random distributions of $\mu,\mu'$ boundary hoppings, while (b) shows the combined energy spectra $E$ (red crosses) from 40 different random trials, which fills up the interior of the elliptical PBC loop of the model given by Eq.\eqref{eq1-2} with parameters $t=2,t'=1,L=50$. From Table I, only the distribution $P(\mu)$ affects the filling, which here forms ``bands'' of approximately equal density due to the chosen step-like distribution of $P(\mu)$. The energies from an illustrative $(\mu,\mu')$ pair is indicated in light blue.
}
\end{figure}

\section{Filling 2D spectral region with a 1D non-Hermitian lattice}

Previously, we filled the 2D spectral region enclosed by the PBC loop via an ensemble of 1D chains. However, to do the same with a \emph{single} 1D chain is a nontrivial feat, even for one constructed by concatenating many 1D chains. This is because the NHSE relentlessly pump all states towards one direction, including across the concatenated chains, thereby fundamentally modifying their individual nature. Specifically, we expect different behavior from boundary hoppings that close up the individual 1D chains, compared to those that connect many 1D chains into one long periodic chain.

To construct a bona fide 1D lattice whose spectrum does fill up a 2D spectral region, we instead consider a model $H$ of the form
\begin{equation}\label{eq2-1}
\begin{split}
H=&\sum^N_{\alpha=1}H_{\alpha}+H_b\ ,\\
\end{split}
\end{equation}
where the first term is that of $N$ Hatano-Nelson chains of length $L$ without boundary hoppings i.e.
\begin{equation}\label{eq2-2}
\begin{split}
&H_{\alpha}=\sum^{L-1}_{n=1} t |\alpha,n\rangle\langle \alpha, n+1|+   t'|\alpha,n+1\rangle\langle \alpha, n|\ ,
\end{split}
\end{equation}
where $\alpha$ is the chain index,  and $n$ labels the sites in the $\alpha$-th chain.
The second term $H_b$ contains all the random couplings between adjacent chains $\alpha$ and $\alpha+1$, and takes the form
 \begin{equation}\label{eq2-3}
H_b=\sum_{\alpha=1}^{N}|\Phi_{\alpha}\rangle\Xi_{\alpha} \langle\Psi_{\alpha+1}|+   |\Psi_{\alpha+1}\rangle\Xi'_{\alpha}\langle \Phi_{\alpha} |  \ ,\\
\end{equation}
where  $|\Phi_{\alpha}\rangle$ and $|\Psi_{\alpha}\rangle$ respectively contains the first $M$ and last $M$ sites of the $\alpha$-th chain i.e.
\begin{equation}\label{eq2-4}\begin{split}
 |\Phi_{\alpha}\rangle&=\left( |\alpha,1\rangle,|\alpha,2\rangle,...,|\alpha,M\rangle\right)\ ,\\
|\Psi_{\alpha}\rangle&=\left( |\alpha,L-M+1\rangle,...,|\alpha,L-1\rangle,|\alpha,L\rangle\right)\ ,\\
\end{split}\end{equation}
and $\Xi_{\alpha},\Xi'_{\alpha}$ are $M\times M$ random matrices whose elements represent their random couplings between the chains. Together, they connect the chains into a long PBC loop via $2NM^2$ random couplings. As elaborated later, the coupling length $M\in [1,L]$ profoundly affects the filling, as demonstrated in Fig.\ref{fig:2}(c) and subsequent figures.

We choose the random couplings matrix elements $\xi\text{e}^{i\phi}$ of $\Xi_{\alpha},\Xi'_{\alpha}$ from a random ensemble with amplitudes $\xi$ Gaussian distributed with mean $0$ and variance $\sigma$, and phase $\phi$ uniformly distributed [Fig.\ref{fig:2}(a)]. Since the spectrum changes dramatically when inter-chain couplings are small, analogous to the boundary couplings discussed in the previous section, the coupling amplitude variance $\sigma$ significantly affects the filling behavior [Fig.\ref{fig:2}(b)].

 \begin{figure}[bp]
    \begin{centering}
    \includegraphics[width=1\linewidth]{figure2.pdf}
    \par\end{centering}
    \protect\caption{\label{fig:2}\textbf{Filling of spectral region from a single long chain with disordered couplings.}
		(a) Structure of our 1D chain model Eqs.(\ref{eq2-1}-\ref{eq2-3}), which consists of $N$ Hatano-Nelson chain segments (green and yellow) that are randomly coupled to adjacent chains via their first and last $M$ sites, as given by matrices $\Xi$ and $\Xi'$. The hopping amplitudes $\xi\in (0, \sigma)$ are Gaussian random distributed, and phases $\phi$ are uniformly distributed. (b) The dependence of the Coverage rate (Cr) Eq.\eqref{eq2-5} and Participation rate  (Pr)  Eq.\eqref{eq2-6} on the hopping amplitude variance $\sigma$, for different $M$. While Cr is generally insensitive to $\sigma$, it improves significantly with $M$. Pr remains almost complete at $1$ for $\sigma\leq 0.1$, beyond which it decreases. (c1-c4) shows the energy spectrum at different $\sigma$ and $M$ combinations as indicated in (b); as $M$ increases, interior eigenvalues within the spectral loop proliferate, finally resulting in a filled spectral interior. Excessive $\sigma$, however, causes the filling to exceed the PBC loop boundaries. (d) The best filling spectrum from (b), with representative eigenstates shown in (e). Generally, more localized states occur deeper in the loop interior. $E_{\text{OBC}}$ and $E_{\text{PBC}}$ refer to the eigenenergies of base model  $H_{\alpha}$ Eq.(\ref{eq2-2}).
}
\end{figure}

To quantify how completely and evenly the spectral loop region is filled by the eigenenergies, we introduce two metrics: Cr, the coverage rate and Pr, the participation rate.
To define the coverage rate (Cr), we divide the interior region of the PBC loop (spectrum of $H_{\alpha}$ under PBCs) into $\mathcal{N} $ parts (for a large $\mathcal{N} $); then we count the number $\mathcal{N}'$ of parts which contain one or more eigenvalues of $H$ Eq.\eqref{eq2-1} within itself. The ratio
\begin{equation}\label{eq2-5}
\begin{split}
\text{Cr}=\frac{\mathcal{N}'}{\mathcal{N}} \ ,
\end{split}
\end{equation}
is the coverage rate Cr. The larger the Cr, the more complete is the energy filling; a small Cr indicates that the filling occurs very inhomogeneously. This definition of Cr remains meaningful even when the PBC loop is irregular and it is hard to directly see how well it is filled by the eigenenergies. Next, we also define the participation rate (Pr), which represents the fraction of eigenenergies of $H$ that are within the PBC loop of $H_\alpha$ i.e.
\begin{equation}\label{eq2-6}
\begin{split}
\text{Pr}=\frac{\text{num}(E)}{NL}\ ,
\end{split}
\end{equation}
where $\text{num}(E) $ is the number of eigenenergies within the PBC loop, and $NL$ is the total number of sites in our lattice model Eq.\eqref{eq2-1}. A high participation rate indicates that few eigenenergies are outside the PBC loop.

From Fig.\ref{fig:2}(b), it is evident that the longer the range $M$ of the inter-chain random couplings, the better the coverage Cr. This is because short-ranged inter-chain couplings in $\Xi_{\alpha},\Xi'_{\alpha}$ only couple sites close to the end of the chains, and the entire lattice is still akin to a long PBC chain with somewhat complicated couplings. Indeed, at small $M$, the spectrum of $H$ is still a well-defined loop [Fig.\ref{fig:2}(c1)], which by definition cover the PBC interior region very poorly. As $M$ increases, more distant sites between adjacent chains are randomly coupled, and some eigenenergies start to appear in the interior of the spectral loop of $H$ [Fig.\ref{fig:2}(c2)]. They almost always appear in its interior because disorder generically break translation invariance, thereby allowing for localized skin mode accumulation. These localized modes have larger effective $\kappa$ by virtue of their shorter decay lengths, and hence tend towards the interior of the spectral loop. This is further examined in the next section; here we mention that intuitively, we expect these random coupling-induced localized skin modes energies to be closer to the real line because the net effect of many random couplings is to prevent any state from being amplified or attenuated too many times consecutively. Finally, the coverage Cr reaches its maximum [Fig.\ref{fig:2}(b)] when $M=L$ i.e. when all sites in our lattice Eqs.(\ref{eq2-1}-\ref{eq2-3}) are randomly coupled. In this limit, all vestiges of a PBC spectral loop are gone, and we observe a continuous eigenstate density within the PBC spectral loop of $H_\alpha$.

The filling of the 2D spectral region can be further optimized by tweaking the probability distribution of the individual inter-chain hopping strengths $\xi e^{i\phi}$. We shall keep the phase $\phi$ as being uniformly distributed, and just vary the variance $\sigma$ of the Gaussian-distributed amplitude $\xi$ (with zero mean). As shown in Fig.\ref{fig:2}(c), the cloud of eigenenergies become larger as $\sigma$ grows, since stronger random couplings invariably perturb their ``trapped'' localized skin modes energies more strongly. We obtain the best filling with $M=L$, and when the eigenenergy cloud just fills the PBC spectral loop of $H_\alpha$ without going out of it. This occur at a critical value of $\sigma=\sigma_{\text{c}}$, where the participation ratio Pr just starts to decrease from 1 [Fig.\ref{fig:2}(b)].

  \begin{figure}[t!]
    \begin{centering}
    \includegraphics[width=1\linewidth]{figure3.pdf}
    \par\end{centering}
    \protect\caption{\label{fig:3}
\textbf{Filling of spectral region from a single long heterogeneous chain with disordered couplings.} (a) Base lattice structure of our model Eq.(\ref{eq3-3}) with next-nearest neighbor hoppings; random hoppings linking them (shown in Fig.\ref{fig:2}) having amplitudes $\xi\in (0, \sigma)$ that are Gaussian random distributed, and phases $\phi$ that are uniformly distributed. (b) Division of the irregularly shaped PBC loop into  large $\mathcal{N}$ parts for the computation of Cr, the Coverage rate. (c) Dependence of the Coverage rate (Cr) Eq.\eqref{eq2-5} and Participation rate  (Pr)  Eq.\eqref{eq2-6} on the hopping amplitude variance $\sigma$, for different $M$. While Cr increases only slightly with $\sigma$, it improves significantly with $M$. Pr remains almost complete at $1$ for $\sigma\leq 0.15$, beyond which it decreases. (c1-c4) shows the spectrum at different $\sigma$ and $M$ combinations as indicated in (c); as $M$ or $\sigma$ increase, interior eigenvalues within the spectral loop proliferate, finally resulting in a filled spectral interior. (d) The best filling spectrum from (c), with representative eigenstates shown in (e). Generally, more localized states occur deeper in the loop interior. $E_{\text{OBC}}$ and $E_{\text{PBC}}$ refer to the eigenenergies of base model  $H_{\alpha}$ Eq.(\ref{eq3-3}).
}
\end{figure}

So far, we have only used the Hatano Nelson model Eq.(\ref{eq2-2}) as the base model $H_{\alpha}$. As a model with a simple elliptical PBC loop, it is an appropriate paradigmatic model for disorder spectral filling. However, our approach also works for generic models with nontrivial spectral winding loops.

For instance, let's add to the disorder coupling $H_b$ Eq.\eqref{eq2-3} a different base Hamiltonian given by~\cite{PhysRevB.102.085151}
 \begin{equation}\label{eq3-3}
\begin{split}
&H_{\alpha}=\sum^{L-1}_{n=1} t |\alpha,n\rangle\langle \alpha, n+a|+   t'|\alpha,n+b\rangle\langle \alpha, n|\ ,
\end{split}
\end{equation}
which has hoppings $a$ sites to the left with amplitude $t$, and hoppings $b$ sites to the right with amplitude $t'$, as illustrated in Fig.\ref{fig:3}(a). These $H_\alpha$ form $N$ identical chains with open boundary conditions, connected to each other by disordered couplings just as before Eqs.(\ref{eq2-1},\ref{eq2-3}).

In the same way, we can also get the best filling conditions. The larger the Cr, the better the spectral region filling, in Fig.\ref{fig:3}(c). There have same results with the previous Hatano-Nelson model,  the value of $M$ is closer to $L$, the better PBC loop-filled energy spectrum.
Similarly,  the model with the best loop-filled energy spectrum can be obtained ($M=L$, and the optimal distribution of random couplings $\xi\exp(i\phi)$ for  Fig.\ref{fig:3}(d) can be obtained by considering the coverage rate (Cr) and participation rate (Pr).

\section{Spectral filling and skin localization}
  \begin{figure}[htb]
    \begin{centering}
    \includegraphics[width=0.95\linewidth]{figure4.pdf}
    \par\end{centering}
    \protect\caption{\label{fig:4} \textbf{Locality (IPR) vs. inverse skin depth $\kappa$.} (a) IPR Eq.\eqref{eq3-1} and (b) $\kappa$ Eq.\eqref{eq3-2} of each eigenstate of our disordered chain Eqs.(\ref{eq2-1}-\ref{eq2-3}) at parameters of best filling ($\sigma=0.1,M=20$). Generally, small $\kappa$ (green), which correspond to delocalized Bloch states in the base model Eq.(\ref{eq2-2}), indeed correlate with relatively delocalized states in the disordered chain (low IPR, green).
(c) and (d) shows the same plots with the base Hamiltonian described by Eq.\eqref{eq3-3}, with similar conclusions.}
\end{figure}

A major inspiration for our approach has been the semi-infinite boundary condition, which is consistent with the coexistence of a continuum of eigenstate decay lengths (skin depths)~\cite{PhysRevLett.125.126402,PhysRevB.104.125109,circuitliu2021non,xiao2020non,PhysRevLett.121.086803,
PhysRevB.99.201103,
PhysRevResearch.1.023013,Longhi:20,li2020critical,PhysRevLett.125.118001,PhysRevLett.124.066602,PhysRevLett.123.066404,PhysRevB.100.035102}. Within our approach, the filling of the spectral loop is indeed intimately related to the realization of a continuum of spatial localization lengths. To quantify localization, we recall the definition of the inverse participation ratio (IPR)~\cite{IPRRevModPhys.80.1355,PhysRevB.103.014201,IPRPhysRevA.95.022117,PhysRevB.100.054301,PhysRevB.97.121401,PhysRevLett.122.237601}
\begin{equation}\label{eq3-1}
\begin{split}
\text{IPR}= \frac{\sum\limits_{\alpha,n}|\psi_{\alpha,n}|^4}{\left(\sum\limits_{\alpha,n}|\psi_{\alpha,n}|^2\right)^2}\ ,
\end{split}
\end{equation}
with $|\Psi\rangle=\sum\limits_{\alpha,n}\psi_{\alpha,n}|\alpha,n \rangle$ a chosen eigenstate of the full Hamiltonian $H$. If a state is perfectly localized on only one site, the IPR takes the maximal value 1. In contrast, if a state is uniformly spread over $NL$ states,  $\text{IPR}=(NL)^{-1}\rightarrow 0$.

We next consider what can be reasonable expectation of the extent of localization for states within the spectral loop. States in a disordered system are invariably randomly shaped, but it is conceivable that they inherit the localization length of the clean background system $H_\alpha$, stochastically speaking. For a particular state with energy $E$, its expected (clean) skin depth $\kappa^{-1}$ is given through
\begin{equation}\label{eq3-2}
\begin{split}
E_{\alpha}(k+i\kappa)=E,
\end{split}
\end{equation}
where $k$ is an unimportant wavenumber and $E_{\alpha}$ is the energy of the 1-component model  $H_{\alpha}$. On the PBC loop, the clean system harbors Bloch states, and $\kappa=0$. Generally, states further from the PBC loop correspond to larger $\kappa$, and should be more localized.

In Fig.\ref{fig:4}, we indeed observe a correlation between weaker expected localization $\kappa$ and weaker numerically determined localization (smaller IPR) of the actual eigenstates. Conversely, for eigenenergies closer to $E_{\text{OBC}}$, where $\kappa\to \kappa_{\text{c}}$,  the eigenstates tend to be most localized (large IPR, red). This is also consistently observed for the base model $H_{\alpha}$ Eq.\eqref{eq3-3} with asymmetric hopping distances, albeit with slightly weaker correlation. \\

\section{Discussion}

In this work, we have devised a way to construct disordered 1D non-Hermitian chains  Eqs.(\ref{eq2-1}-\ref{eq2-3}) exhibiting eigenspectra that fill up the interiors of 2D regions in the complex energy plane. The filling extent and density can be adjusted by tuning the probability distributions of the random couplings, and effectively simulates the effects of SIBCs, which are physically unattainable.

It is interesting to compare our mechanism with that of random matrices i.e. the Gaussian unitary, orthogonal and symplectic ensembles etc, which can also produce evenly spaced eigenvalues within circular regions in the complex plane, akin to electrons in a quantum Hall fluid~\cite{frohlich1991large,PhysRevB.52.14137,PhysRevLett.75.697,susskind2001quantum,PhysRevLett.95.176402,Bergman_2001,jokela2011magneto,PhysRevB.90.115139,PhysRevB.99.035427}. What is markedly different is the extent of non-locality required: In our setup [Fig.\ref{fig:2}(a)], random inter-segment couplings extend across at most $M$ sites, and in the thermodynamic limit of large $N$, the entire chain can still be considered as a long PBC chain with $N$ nearest-neighbor coupled unit cells, each having a fixed $L\geq M$ number of components. However, in classical random matrix ensembles, the random elements represent all possible couplings, which in this context can be as far as $NL/2$ number of sites. Furthermore, our approach can easily be generalized to fill up arbitrarily-shaped regions by choosing the base Hamiltonian $H_\alpha$ with a similarly shaped PBC loop.

From a more general viewpoint, our constructive approach offers an avenue for stochastically ``augmenting'' the dimensionality of a 1D system, such that it possess characteristics normally associated with 2D systems, such as 2D density of states. The probability distributions associated with the random couplings provide additional degrees of freedom that may ultimately emulate extra dimensions. Finally, we mention that our models can readily be physically implemented in media tha admit long-ranged couplings, such as classical electrical circuits~\cite{circuitPhysRevLett.114.173902,circuitSTP,circuitLee2018,circuitimhof2018topolectrical,circuitPhysRevB.99.161114,circuitPhysRevLett.122.247702,circuitPhysRevB.100.081401,circuitPhysRevB.99.020302, circuitPhysRevB.100.184202,experimenthelbig2020,lee2020imaging,circuitPhysRevLett.124.046401,circuithofmann2020reciprocal,lenggenhager2021electric,circuitliu2021non,circuitrafi2021non,circuitzou2021observation,zhang2021observation,circuitPhysRevResearch.3.023056,circuitPhysRevLett.126.215302,circuitliu202001,RevModPhys.93.015005}
 and quantum computers~\cite{garcia2020ibm,ippoliti2021many,choo2018measurement,smith2019simulating,helsen2019spectral,randall2021many,frey2021simulating,stenger2021simulating,koh2021stabilizing,experimentwu2019,PhysRevX.10.021054,PhysRevX.8.011032,madjarov2020high},
 and with some adaptation even Rydberg atom lattices with long-ranged interactions~\cite{PhysRevX.10.021054,PhysRevX.8.011032,madjarov2020high,PhysRevLett.104.173001,anderson2011trapping,PhysRevA.86.053618,bluvstein2021controlling,keesling2019quantum,omran2019generation,PhysRevLett.124.103601,liu2020localization,samajdar2021quantum,PhysRevResearch.3.043059}. Since the non-local couplings only need to be randomly distributed according to certain loosely defined distributions, our approach is intrinsically tolerant to significant levels of noise.


I have always been a giant fan of 20th century American music. There are so many wonderful composers to choose from whose music works well in an outdoor setting. Gershwin, Bernstein and Sousa are all amazing, but for me, Aaron Copland's music is the best. I have always been fascinated with his ability to capture the essence of our American spirit. So when I sat down to think about this summer's program, I knew that Copland needed to play a major role. It didn't take me too long to settle on "Americana" as this year's theme. When you come to hear us this summer, we will be taking you on a historical journey of American music. We will be performing pieces by masters such as Charles Ives, John Williams and John Philip Sousa juxtaposed against new and exciting composers like Michael Markowski and Jim Territo. Of course, Aaron Copland will make an appearance, too.
The summer will be here soon and I hope you will enjoy this program as much as I have enjoyed putting it together. I cannot wait to get started!


text:
Q: Microsoft AJAX with debugger There is a couple of problems which is related to microsoft ajax and visual studio when it is attached to w3wp. When i run my site when Visual studio hasn't been attached to it microsoft ajax works fine. But when i have attached to VS and use site again it rises some exceptions in DevTools js console.
An error occured :

Update - when an error is occured v4 equals false, but when all work fine v4 equals to true.

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Home » Blog » Newsflash! The Supreme Court delivers Judgement in Owens vs. Owens: trapped in marriage?
Newsflash! The Supreme Court delivers Judgement in Owens vs. Owens: trapped in marriage?
Mrs Owens appealed against an order of the Court of Appeal of 24th March 2017 by which it dismissed her appeal against the dismissal of the petition for divorce. The Supreme Court heard the matter on 17th May 2018.
Mrs Owens and Mr Owens married in 1978, they are aged 68 and 80 respectively. During the marriage, Mr Owens started a company which has made both parties significantly wealthy. Mr Owens continues to reside in the former matrimonial home, a manor house in Gloucestershire. Mrs Owens resides next door, in a property which they also own.
Mrs Owens first petitioned for divorce in May 2015 relying on section 1(2)(b) of the Matrimonial Causes Act 1973, that her husband had behaved in such a way that that she cannot reasonably be expected to live with him. This was 3 years after first consulting solicitors about a divorce and having sent a draft petition to Mr Owen's solicitors, all of which occurred whilst she was having an affair which subsequently ended in May 2013. Mrs Owens explained to the judge that the affair was as a result of a "bad marriage", not the cause of the divorce. However the judge suggested that the affair gave her an ulterior motive for the proposed petition, and believed that the existence of her affair "knocks out" the allegations made against Mr Owens in her behaviour based divorce. The Supreme Court disagreed with this view.
Mrs Owens described Mr Owens as having prioritised his work over their family life, his treatment of her lacked love or affection; he had often been moody and argumentative and had disparaged her even in front of others. As a result of his behaviour she felt unhappy, unappreciated, upset and embarrassed and thus has grown apart from him.
As a result of Mr Owens defence of the divorce, Mrs Owens was granted permission to amend her petition so as to provide more details as to the behaviour she intended to rely- which she did, with 27 individual examples. The Judge, however, regarded her examples as "at best flimsy" and criticised her for having over exaggerated their context and seriousness which, they deem was the result of her over-sensitivity.
This case has raised issues about the interpretation of the subsection, as to whether the behaviour itself must constitute sufficient grounds for divorce or whether the emphasis must lie on the effect that the behaviour had on the proposed petitioner. As in this case, Mrs Owens argued that the "entire focus should be on the reaction of the petitioner to the respondent's behaviour" rather than an objective approach of whether the behaviour was sufficient.
The Supreme Court were persuaded to dismiss the appeal of Mrs Owens, on the premise that if they were to allow the appeal and send the case back to be re-tried, it would be contrary to the interests of the parties and such a prospect would fill the petitioner with, in her own words, "dread". Consideration was given to the fact that in February 2020, the parties would have been separated for 5 years and thus, Mrs Owens would be entitled to petition for a divorce on the ground of 5 years separation without consent. For now, therefore, Mrs Owens remains married to Mr Owens in law.
This case teaches us that the court's will have the power to strike out a petition for divorce in the case that the grounds upon which the petitioner intends to rely are not sufficient.
The law surrounding divorce is, at best, outdated. Whilst there remains a requirement of fault and blame which often inflames the relationship, there is no room for amicable separation even though it is recorded that "agreed outcomes are less costly and damaging, [and] that trying to apportion blame is a fruitless and inherently non-justiciable task and that defence is futile where one party has decided the marriage is over." There has been much momentum in the 'no fault divorce' which will allow parties of a divorce to do so immediately i.e. not waiting for two years separation with consent, and without the need for extensive justification. We will keep up to date in the progression of the reform.


As part of the Cha Writing Workshop Series, in partnership with the Hong Kong Poetry Festival Foundation and supported by the English Departments at CUHK and HKBU, I have now taught nine 90-minute creative writing workshops for female refugees affiliated with the Christian Action Centre for Refugees.
I have used poems by Maya Angelou, George Ella Lyon, Ghalib, Darwish, Renato Resaldo and Aaliya Zaveri, among others. The lessons begin with listening to the poet reading his or her poem, if possible. Then we read the poem together (usually one of the more advanced students does that willingly). After that, we discuss idioms, slang terms and concepts from the poems, so the participants can share their knowledge with each other. Then the group does vocabulary and picture matching to find the meanings of more concrete nouns and verbs. After taking up the answers, we look for metaphors and similes and discuss various possible interpretations. Finally, the students use a close version of the poem with missing keywords to help them create a poem of their own "after" the model poet of the day.
I am learning a lot as I teach this group about how to tutor them more effectively. Listening, repeating back and other trust-building responses help foster a safe and comfortable learning environment. Women who attend have IELTS levels from 4 to 8 in writing, yet the group's spirit of cooperation and mutual support means there is a lot of peer learning.
I would say these workshops with female refugees and the workshops I have begun with the group "Minds at Wok" (with domestic helpers in Hong Kong) are the most rewarding I have developed and taught in several years. As grade inflation at college and even university becomes the norm and teaching at those institutions becomes more and more about satisfying students and their parents as "customers," and, as students at those levels read less and less, it is very meaningful to teach students who are motivated and eager to learn. I am grateful for the opportunity and very glad to give back through my teaching to people who are often relegated to the margins of society.
This entry was posted in Cha Writing Workshop Series, Christian Action Centre for Refugees, CWWS_Kate Rogers, Kate Rogers, Minds at Wok. Bookmark the permalink.


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