Mackey-Borel structure

A Borel structure (i.e., a Borel system of subsets) on the spectrum $ \widehat{A} $ of a separable $ C^{*} $-algebra $ A $ (cf. also Spectrum of a $ C^{*} $-algebra), defined as follows. Let $ \mathcal{H}_{n} $, where $ n \in \mathbb{N} $, be a Hilbert space of dimension $ n $, and let $ {\operatorname{Irr}_{n}}(A) $ denote the set of non-zero irreducible representations (cf. Irreducible representation) of $ A $ on $ \mathcal{H}_{n} $ equipped with the topology of pointwise convergence in the weak topology. Let on $ {\operatorname{Irr}_{n}}(A) $ be given the Borel structure generated by its topology (i.e., the smallest Borel structure relative to which all mappings $ \pi \mapsto \langle [\pi(x)](\xi),\eta \rangle $ — where $ x \in A $, $ \xi,\eta \in \mathcal{H}_{n} $ and $ \pi \in {\operatorname{Irr}_{n}}(A) $ — are Borel functions), and let $ \operatorname{Irr}(A) $ denote the union of the sub-spaces $ {\operatorname{Irr}_{n}}(A) $, $ n \in \mathbb{N} $, provided with the Borel structure such that a subset of $ \operatorname{Irr}(A) $ is a Borel set if and only if its intersection with each $ {\operatorname{Irr}_{n}}(A) $ belongs to the Borel structure on the latter. Let $ \phi $ denote the mapping of the Borel space $ \operatorname{Irr}(A) $ into the spectrum $ \widehat{A} $ of $ A $ that maps a representation to its unitary equivalence class. The Borel structure on $ \widehat{A} $ generated by the sets whose inverse images under $ \phi $ are Borel sets in $ \operatorname{Irr}(A) $ is called the Mackey–Borel structure on $ \widehat{A} $. The Mackey–Borel structure contains all sets of the Borel structure generated by the topology of $ \widehat{A} $; each point of $ \widehat{A} $ is a Borel set in the Mackey–Borel structure. The following four conditions are equivalent:

1. The Mackey–Borel structure is standard (i.e., it is isomorphic, as a Borel structure, to the Borel structure generated by the topology of some complete separable metric space).
2. The Mackey–Borel structure coincides with the Borel structure generated by the topology on $ \widehat{A} $.
3. The Mackey–Borel structure on $ \widehat{A} $ is countably separated.
4. If $ A $ is a $ \mathsf{GCR} $-algebra, then a Mackey–Borel structure can also be introduced on the quasi-spectrum of a separable $ C^{*} $-algebra.

References

[1]	J. Dixmier, “$ C^{*} $-algebras”, North-Holland (1977). (Translated from French)
[2]	L.T. Gardner, “On the Mackey Borel structure”, Canad. J. Math., 23: 4 (1971), pp. 674–678.
[3]	H. Halpern, “Mackey Borel structure for the quasi-dual of a separable $ C^{*} $-algebra”, Canad. J. Math., 26: 3 (1974), pp. 621–628.

References

[a1]	W. Arveson, “An invitation to $ C^{*} $-algebras”, Springer (1976), Chapts. 3–4.