Riggings of locally compact abelian groups
RIGGINGS OF LOCALLY COMPACT
ABELIAN GROUPS.
M. Gadella, F. Gómez, S. Wickramasekara.
Abstract
We obtain a set of generalized eigenvectors that provides a gen-
eralized spectral decomposition for a given unitary representation of
a commutative, locally compact topological group. These generalized
eigenvectors are functionals belonging to the dual space of a rigging on
the space of square integrable functions on the character group. These
riggings are obtained through suitable spectral measure spaces.
1 Introduction
The purpose of the present paper is to take a first step towards a general
formalism of unitary representations of groups and semigroups on rigged
Hilbert spaces. To begin with, we want to introduce the theory correspond-
ing to Abelian locally compact groups, leaving the more general nonabelian
case as well as semigroups for a later work. We recall that a rigged Hilbert
space or a rigging of a Hilbert space H is a triplet of the form
Φ ⊂ H ⊂ Φ× , (1)
where Φ is a locally convex space dense in H with a topology stronger than
that inherited from H and Φ× is the dual space of Φ. In this paper, we
shall always assume that H is separable.
To each self adjoint operator A on H, the von Neumann theorem [1]
associates a spectral measure space. This is the quadruple (Λ,A,H, P ),
whereH is the Hilbert space on which A acts, Λ = σ(A) is the spectrum of A,
A is the family of Borel sets in Λ, and P is the projection valued measure on
A determined by A through the von Neumann theorem. Obviously Λ ⊂ R.
A complete discussion on the relation between these concepts can be found
in [2]. We say that the topological vector space (Φ, τΦ) (vector space Φ with
http://arxiv.org/abs/0704.1634v1
the locally convex topology given by τΦ) equips or rigs the spectral measure
(Λ,A,H, P ) if the following conditions hold:
i. There exists a one-to-one linear mapping I : Φ 7−→ H with range
dense in H. We can assume that Φ ⊂ H is a dense subspace of H and
I, the canonical injection from Φ into H.
ii. There exists a σ-finite measure µ on (Λ,A), a set Λ0 ⊂ Λ with zero µ
measure and a family of vectors in Φ× of the form
{|λk×〉 ∈ Φ× : λ ∈ Λ\Λ0, k ∈ {1, 2, . . . ,m}}, (2)
where m ∈ {∞, 1, 2, . . .}, such that
(φ, P (E)ϕ)H =
〈φ|λk×〉 〈ϕ|λk×〉∗ dµ(λ), ∀φ,ϕ ∈ Φ, ∀E ∈ A.
Each family of the form (2) satisfying (3) is called a complete system of
Dirac kets of the spectral measure (Λ,A,H, P ) in (Φ, τΦ). In this case, the
triplet Φ ⊂ H ⊂ Φ× is a rigged Hilbert space, which is called a rigging of
(Λ,A,H, P ).
Conversely, the von Neumann theorem asserts that a projection valued
measure defined on the σ-algebra of Borel sets on a subset of the real line
determines a self adjoint operator A. If (Λ,A,H, P ), where Λ ⊂ R, is such a
measure space, then for ϕ and φ on a suitable dense domain, the self-adjoint
operator A such that Λ = σ(A) is defined by
(φ,Aϕ)H =
λ 〈φ|λk×〉 〈ϕ|λk×〉∗ dµ(λ) (4)
where µ, |λk×〉 and m are as defined in (3). Further, if f(λ) is a measurable
complex valued function on Λ, then, for φ,ϕ on a suitable dense domain,
which is the whole of H if f(λ) is bounded, the operator valued function
f(A) is defined by
(φ, f(A)ϕ)H =
f(λ) 〈φ|λk×〉 〈ϕ|λk×〉∗ dµ(λ) . (5)
The functionals |λk×〉 ∈ Φ× and the complex numbers f(λ) are the gen-
eralized eigenvectors and respective generalized eigenvalues of f(A) [3]. In
particular, if f(λ) = eitλ, where t ∈ R, the set of operators eitA forms a one
parameter commutative group of unitary operators and Φ ⊂ H ⊂ Φ× as
defined above is a rigging for this group.
One can expect that similar riggings exist for unitary representations of
arbitrary groups and semigroups and that the operators of the representa-
tions can be expanded in terms of generalized eigenvectors and eigenvalues
as in (5). Riggings that make use of Hardy functions on a half plane exist
for one parameter dynamical semigroups e−itH , t ≤ 0 and e−itH , t ≥ 0,
where H is the Hamiltonian [3].
In the present paper, we show that riggings along the above lines al-
ways exist for unitary representations of Abelian locally compact groups.
In particular, let G be an Abelian locally compact group and π, a unitary
representation of G on a separable Hilbert space H. We will see that the
Fourier transform on G, or equivalently, the Gelfand transformation on the
C∗-algebra L1(G) allows us to represent π in terms of generalized eigenfunc-
tions and riggings of H in a manner similar to the description given in [2]
for the action of a spectral measure.
2 Characters of Abelian Locally Compact Groups.
Let G be a locally compact abelian group with Haar measure µ. A character
χ of G is any continuous mapping from G into the set of complex numbers
C such that χ(g1g2) = χ(g1)χ(g2) for all g1, g2 ∈ G and |χ(g)| = 1 for all
g ∈ G, i.e., a character of G is a continuous homomorphism from G into the
unit circle T. The set of all the characters of G forms a group, Ĝ, which is
often called the dual group of G. We shall use the notation χ(g) := 〈g|χ〉.
Let L1(G) be the space of complex valued functions, integrable in the
modulus with respect to the Haar measure µ on G. L1(G) is an abelian ∗-
algebra, with the convolution product. The dual group Ĝ can be identified
with the set of maximal ideals of L1(G) [4]. When endowed with the Gelfand
topology, Ĝ is a compact Hausdorf space (see [5] page 268).
For any χ ∈ Ĝ, we may define a linear functional Λχ on L
1(G) by
Λχ(f) =
〈g|χ〉∗f(g)dµ(g) . (6)
Let C(Ĝ) be the space of complex continuous functions on Ĝ with the
supremun norm topology. The Gelfand-Fourier transform is the mapping
F : L1(G) 7−→ C(Ĝ) defined by:
[Ff ](χ) = f̂(χ) = Λχ(f) =
〈g|χ〉∗ f(g) dµ(g) . (7)
Let (π,H) be a unitary representation of G. Then (see [6] page 105),
there is a unique spectral measure (Ĝ,B,H, P ), where B is the σ-algebra of
Borel sets on Ĝ, such that for all g ∈ G and all f ∈ L1(G), we have
π(g) =
〈g|χ〉 dP (χ) ; π(f) =
Λχ(f) dP (χ) . (8)
There is a one to one correspondence between unitary representations of G
and non degenerate ∗-representations1 of L1(G) as given by (7) and (8).
2.1 Riggings of functions of characters.
Let us consider the spectral measure space (Ĝ,B,H, P ) introduced in the
previous section. For simplicity in the discussion, we assume the existence
of a cyclic vector u ∈ H. This means that the subspace spanned by the
vectors of the form P (E)u with E ∈ A is dense in H. The general case can
be easily obtained as a finite or countable direct sum of cyclic subspaces of
Then, the von Neumann decomposition theorem [1] establishes that be-
ing given the spectral measure space (Ĝ,B,H, P ) and a positive measure
ν on (Ĝ,B) with maximal spectral type2 [P ] (ν ∈ [P ]), there exists a uni-
tary mapping U : H 7−→ L2(Ĝ, dµ) , such that πν(g) := Uπ(g)U
−1 is the
multiplication by 〈g|χ〉 on L2(Ĝ, dν):
πν(g)φ(χ) = Uπ(g)U
−1φ(χ) = 〈g|χ〉φ(χ) , ∀φ(χ) ∈ L2(Ĝ, dν) . (9)
Since πν(g) is a multiplication operator, it is easy to see that the Dirac
delta type Radon measures λ(χ)δχ form a complete system of Dirac kets for
the spectral measure space (Ĝ,B,H, P ) in the sense given by (3). For any
f(χ) ∈ Φ these deltas satisfy
f(χ) δχ′ dν = f(χ
′) . (10)
Thus, a possible choice for Φ is C(Ĝ), the space of continuous functions
on Ĝ endowed with a topology τΦ stronger than both the topologies of the
supremun and the || · ||
L2( bG,dν)
norm. In this case, the dual Φ× of Φ includes
the space of all Radon measures on (Ĝ,B). We have the rigged Hilbert space
Φ ⊂ L2(Ĝ) ⊂ Φ×.
1Non degenerate means that π(f)v = 0 for every f implies v = 0. The representation
has also the property that π(f∗) = π†(f), where f 7→ f∗ is the involution on L1(G), see
2For a definition and properties of the spectral type, see [1, 2].
3 Positive Type Functions and Riggings.
Next, we shall introduce another representation πφ of G linked to a function
of positive type, that can be defined as follows: Let φ(g) ∈ L∞(G). We say
that φ(g) is a function of positive type if for any f(g) ∈ L1(G), we have that
f∗(g)f(gg′)φ(g′) dµ(g) dµ(g′) ≥ 0 , (11)
where the star ∗ denotes complex conjugation.
If φ(g) is a function of positive type, then, the following positive Hermi-
tian form on L1(G)
〈h|f〉φ :=
h∗(g′)f(g)φ(g−1g′) dµ(g′) dµ(g) (12)
is semi-definite in the sense that it may exist non-zero functions f ∈ L1(G)
such that 〈f |f〉φ = 0. These functions form a subspace of L
1(G) that we
denote by N . Consider the factor space L1(G)/N and again denote by 〈·|·〉φ
the scalar product induced on L1(G)/N by the Hermitian form (12). The
completion of L1(G)/N by 〈·|·〉φ gives a Hilbert space usually denoted as
Hφ. Then, for any g ∈ G and f(g) ∈ L
1(G), we define:
(Lg)f(g
′) := f(g−1g′) . (13)
Note that Lg preserves the scalar product 〈·|·〉φ:
〈Lgh|Lgf〉φ =
h∗(g−1g′)f(g−1g′′)φ(g
−1g′′) dµ(g′) dµ(g′′)
h∗(g′)f(g′′)φ((gg′)−1(gg′′)) dµ(g′) dµ(g′′) = 〈h|f〉φ , (14)
for all f(g) ∈ L1(G). This also shows that LgN ⊂ N and therefore Lg
induces a transformation on the factor space L1(G)/N , that we also denote
as Lg, defined as
Lg(f(g
′) +N ) := f(g−1g′) +N = Lg(f(g
′)) +N . (15)
By (14), we easily see that Lg preserves the scalar product on L
1(G)/N . It
is obviously invertible. Therefore, it can be uniquely extended into a unitary
operator on Hφ. Then, if for each g ∈ G we write
πφ(g)f := Lgf , ∀ f ∈ Hφ , (16)
then, πφ(g) determines a unitary representation of G on Hφ. The proof of
this statement is straightforward.
The representation πφ(g) of G on Hφ can be lifted to a unitary repre-
sentation of the group algebra L1(G) on Hφ that we shall also denote as πφ.
In this case, for all f ∈ L1(G), we have πφ(f)h := f ∗ h. Here, ∗ denotes
convolution.
The existence of a cyclic vector η ∈ Hφ for the representation πφ is proven
in [6]. Recall that η is cyclic vector if the subspace {πφ(f)η, ∃f ∈ L
1(G)}
is dense in Hφ. In addition, this result also gives the following formula that
allows to find the function φ(g) in terms of η and the unitary representation
πφ of G on Hφ:
φ(g) = 〈η|πφ(g)η〉 . (17)
Now, let us consider the unitary πν representation of G given by (9) with
cyclic vector ξ and define the following complex valued function on G:
φ(g−1g′) := 〈ξ|πν(g
−1g′)ξ〉
L2( bG,dν)
= 〈πν(g)ξ|πν(g
L2( bG,dν)
. (18)
Then, as shown in [6], Chapter 3,
i.) the function φ is of positive type in the sense of (11), and
ii.) the representation ofG onHφ given by πφ, where φ is as (18) is equivalent
to πν .
Note that this result implies in particular that for this φ as in (18)
φ(g) = 〈η|πφ(g)η〉φ = 〈ξ|πν(g)ξ〉L2( bG,dν) , ∀ g ∈ G . (19)
According to (9) and (18), we have that
φ(g−1g′) = 〈πν(g)ξ|πν(g
L2( bG,dν)
[〈g|χ〉 ξ(χ)]∗ 〈g′|χ〉 ξ(χ) dν(χ) .
If we carry this formula into (12) and apply the Fubini theorem of the change
of the order of integration, we have for all f, h ∈ L1(G):
〈f |h〉φ =
[f(g)〈g|χ〉]∗ dµ(g)
h(g′)〈g′|χ〉 dµ(g′)
|ξ(χ)|2 dν(χ)
[f̂(χ)]∗ ĥ(χ) |ξ(χ)|2 dν(χ) =
〈f̂ |χ〉〈χ|ĥ〉 |ξ(χ)|2 dν(χ) . (21)
This latter formula shows that the generalized eigenvalues Fχ of πφ(g) are
the following: if f ∈ Φ := L1(G) ∩ L2(G)
|Fχ〉 ≡ Fχ : f 7−→ |η(χ)|f̂
∗(χ) = |η(χ)|
〈g|χ〉f∗(g) dµ(g) . (22)
We endow Φ with any topology stronger than the topologies L1(G) and
L2(G). For instance, we can choose a locally convex topology with the semi-
norms p1(f) := ||f ||L1(G) and p2(f) := ||f ||L2(G), for all f ∈ Φ. With this
topology or another stronger one, the antilinear functional Fχ is continuous.
Then, if we use (8) in the scalar product on Hφ, we have:
〈f |πφ(g)h〉φ =
〈g|χ〉 d〈f |P (χ)h〉φ =
〈g|χ〉 |η(χ)|2 f̂∗(χ)ĝ(χ) dν(χ)
〈g|χ〉 〈f |Fχ〉〈Fχ|h〉 dν(χ) . (23)
If we omit the arbitrary f, h ∈ Φ in (23), we have the following spectral
decomposition for πφ(g) for all g ∈ G:
πφ(g) =
〈χ|g〉 |Fχ〉〈Fχ| dν(χ) . (24)
Note that in the antidual space Φ×, the generalized eigenvalue equation
πφ(g)|Fχ〉 = 〈χ|g〉|Fχ〉 is valid, where we use the same notation πφ(g) for
the extensions of these unitary operators into Φ×.
In conclusion, for each unitary representation of a locally compact Abelian
topological group, we have found an equivalent representation and a rigged
Hilbert space such that each of the unitary operators of the representation
admits a generalized spectral decomposition in terms of generalized eigen-
vectors of them. The eigenvectors of the decomposition are labeled by the
group characters only and their respective eigenvalues, complex numbers
with modulus one, depend on both the corresponding character and the
group element. The spectral decomposition and the corresponding rigging
comes after the existence of a spectral measure space.
Note that the Abelian property is crucial in our derivation and in partic-
ular in the existence of the spectral measure space (Ĝ,B,H, P ), since then,
the group algebra is also Abelian and the Gelfand theory applies. An ex-
tension of the present formalism to nonabelian locally compact groups will
require an extension of the Gelfand formalism that at least allows for a new
and consistent definition of the Gelfand Fourier transform (7), an essential
feature of our construction.
Acknowledgements
We acknowledge the financial support from the Junta de Castilla y León
Project VA013C05 and the Ministry of Education and Science of Spain,
projects MTM2005-09183 and FIS2005-03988. S.W. acknowledges the fi-
nancial support from the University of Valladolid where he was a visitor
while this work was done and additional financial support from Grinnell
College.
References
[1] J. von Neumann, Mathematical Foundations of Quantum Mechanics
(Princeton University, Princeton, N.J., 1955).
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Gadella, F. Gómez, International Journal of Theoretical Physiscs,
42, 2225-2254 (2003); M. Gadella, F Gómez, Bulletin des Sciences
Mathèmatiques, 129, 567 (2005); M. Gadella, F. Gómez, Reports on
Mathematical Physics, 59, 127 (2007).
[3] A. Bohm, M. Gadella, Dirac kets, Gamow vectors and Gelfand triplets,
Springer Lecture Notes in Physics, 348 (Springer, Berlin 1989).
[4] M.A. Naimark, Normed Rings (Wolters-Noordhoff, Groningen, The
Netherlands, 1970).
[5] W. Rudin, Functional Analysis (McGraw-Hill, New York 1973).
[6] G. B. Folland, A Course in Abstract Harmonic Analysis (CRC, Boca
Raton, London, 1995).
M. Gadella
Departamento de F́ısica Teórica
Facultad de Ciencias
c. Real de Burgos, s.n.
47011 Valladolid, Spain
E-mail address:
manuelgadella@yahoo.com.ar
F. Gómez
Departamento de Análisis Matemático
Facultad de Ciencias
c. Real de Burgos, s.n.
47011 Valladolid, Spain
E-mail address:
fgcubill@am.uva.es
S. Wickramasekara
Department of Physics,
Grinnell College, Grinnell, IA 50112, USA
E-mail address:
WICKRAMA@Grinnell.EDU
	Characters of Abelian Locally Compact Groups.
	Riggings of functions of characters.
	Positive Type Functions and Riggings.
