Spectrum of a C*-algebra

In mathematics, the spectrum of a C*-algebra or dual of a C*-algebra "A", denoted "Â", is the set of unitary equivalence classes of irreducible *-representations of "A". A *-representation π of "A" on a Hilbert space "H" is irreducible if, and only if, there is no closed subspace "K" different from "H" and {0} which is invariant under all operators π("x") with "x" ∈ "A". We implicitly assume that irreducible representation means "non-null" irreducible representation, thus excluding trivial (i.e. identically 0) representations on one-dimensional spaces. As explained below, the spectrum "Â" is also naturally a topological space; this is similar to the notion of the spectrum of a ring.
One of the most important applications of this concept is to provide a notion of dual object for any locally compact group. This dual object is suitable for formulating a Fourier transform and a Plancherel theorem for unimodular separable locally compact groups of type I and a decomposition theorem for arbitrary representations of separable locally compact groups of type I. The resulting duality theory for locally compact groups is however much weaker than the Tannaka–Krein duality theory for compact topological groups or Pontryagin duality for locally compact "abelian" groups, both of which are complete invariants. That the dual is not a complete invariant is easily seen as the dual of any finite-dimensional full matrix algebra M"n"(C) consists of a single point.
Primitive spectrum.
The topology of "Â" can be defined in several equivalent ways. We first define it in terms of the primitive spectrum .
The primitive spectrum of "A" is the set of primitive ideals Prim("A") of "A", where a primitive ideal is the kernel of an irreducible *-representation. The set of primitive ideals is a topological space with the hull-kernel topology (or Jacobson topology). This is defined as follows: If "X" is a set of primitive ideals, its hull-kernel closure is
Hull-kernel closure is easily shown to be an idempotent operation, that is
and it can be shown to satisfy the Kuratowski closure axioms. As a consequence, it can be shown that there is a unique topology τ on Prim("A") such that the closure of a set "X" with respect to τ is identical to the hull-kernel closure of "X".
Since unitarily equivalent representations have the same kernel, the map π ↦ ker(π) factors through a surjective map
We use the map "k" to define the topology on "Â" as follows:
Definition. The open sets of "Â" are inverse images "k"−1("U") of open subsets "U" of Prim("A"). This is indeed a topology.
The hull-kernel topology is an analogue for non-commutative rings of the Zariski topology for commutative rings.
The topology on "Â" induced from the hull-kernel topology has other characterizations in terms of states of "A".
Examples.
Commutative C*-algebras.
The spectrum of a commutative C*-algebra "A" coincides with the Gelfand dual of "A" (not to be confused with the dual "A"' of the Banach space "A"). In particular, suppose "X" is a compact Hausdorff space. Then there is a natural homeomorphism
This mapping is defined by
I("x") is a closed maximal ideal in C("X") so is in fact primitive. For details of the proof, see the Dixmier reference. For a commutative C*-algebra,
The C*-algebra of bounded operators.
Let "H" be a separable infinite-dimensional Hilbert space. "L"("H") has two norm-closed *-ideals: "I"0 = {0} and the ideal "K" = "K"("H") of compact operators. Thus as a set, Prim("L"("H")) = {"I"0, "K"}. Now
Thus Prim("L"("H")) is a non-Hausdorff space.
The spectrum of "L"("H") on the other hand is much larger. There are many inequivalent irreducible representations with kernel "K"("H") or with kernel {0}.
Finite-dimensional C*-algebras.
Suppose "A" is a finite-dimensional C*-algebra. It is known "A" is isomorphic to a finite direct sum of full matrix algebras:
where min("A") are the minimal central projections of "A". The spectrum of "A" is canonically isomorphic to min("A") with the discrete topology. For finite-dimensional C*-algebras, we also have the isomorphism
Other characterizations of the spectrum.
The hull-kernel topology is easy to describe abstractly, but in practice for C*-algebras associated to locally compact topological groups, other characterizations of the topology on the spectrum in terms of positive definite functions are desirable.
In fact, the topology on "Â" is intimately connected with the concept of weak containment of representations as is shown by the following:
The second condition means exactly that π is weakly contained in "S".
The GNS construction is a recipe for associating states of a C*-algebra "A" to representations of "A". By one of the basic theorems associated to the GNS construction, a state "f" is pure if and only if the associated representation π"f" is irreducible. Moreover, the mapping κ : PureState("A") → "Â" defined by "f" ↦ π"f" is a surjective map.
From the previous theorem one can easily prove the following;
The space Irr"n"("A").
There is yet another characterization of the topology on "Â" which arises by considering the space of representations as a topological space with an appropriate pointwise convergence topology. More precisely, let "n" be a cardinal number and let "Hn" be the canonical Hilbert space of dimension "n".
Irr"n"("A") is the space of irreducible *-representations of "A" on "Hn" with the point-weak topology. In terms of convergence of nets, this topology is defined by π"i" → π; if and only if
It turns out that this topology on Irr"n"("A") is the same as the point-strong topology, i.e. π"i" → π if and only if
Remark. The piecing together of the various "Ân" can be quite complicated.
Mackey–Borel structure.
"Â" is a topological space and thus can also be regarded as a Borel space. A famous conjecture of G. Mackey proposed that a "separable" locally compact group is of type I if and only if the Borel space is standard, i.e. is isomorphic (in the category of Borel spaces) to the underlying Borel space of a complete separable metric space. Mackey called Borel spaces with this property smooth. This conjecture was proved by James Glimm for separable C*-algebras in the 1961 paper listed in the references below.
Definition. A non-degenerate *-representation π of a separable C*-algebra "A" is a factor representation if and only if the center of the von Neumann algebra generated by π("A") is one-dimensional. A C*-algebra "A" is of type I if and only if any separable factor representation of "A" is a finite or countable multiple of an irreducible one.
Examples of separable locally compact groups "G" such that C*("G") is of type I are connected (real) nilpotent Lie groups and connected real semi-simple Lie groups. Thus the Heisenberg groups are all of type I. Compact and abelian groups are also of type I.
The result implies a far-reaching generalization of the structure of representations of separable type I C*-algebras and correspondingly of separable locally compact groups of type I.
Algebraic primitive spectra.
Since a C*-algebra "A" is a ring, we can also consider the set of primitive ideals of "A", where "A" is regarded algebraically. For a ring an ideal is primitive if and only if it is the annihilator of a simple module. It turns out that for a C*-algebra "A", an ideal is algebraically primitive if and only if it is primitive in the sense defined above.
This is the Corollary of Theorem 2.9.5 of the Dixmier reference.
If "G" is a locally compact group, the topology on dual space of the group C*-algebra C*("G") of "G" is called the Fell topology, named after J. M. G. Fell.