View source for Mackey-Borel structure

A Borel structure (i.e., a [[Borel system of sets|Borel system of subsets]]) on the spectrum $ \widehat{A} $ of a separable [[C*-algebra|$ C^{*} $-algebra]] $ A $ (cf. also [[Spectrum of a C*-algebra|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|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:

# 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).
# The Mackey–Borel structure coincides with the Borel structure generated by the topology on $ \widehat{A} $.
# The Mackey–Borel structure on $ \widehat{A} $ is countably separated.
# 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====

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<TR><TD valign="top">[1]</TD><TD valign="top"> J. Dixmier, “$ C^{*} $-algebras”, North-Holland (1977). (Translated from French)</TD></TR>
<TR><TD valign="top">[2]</TD><TD valign="top"> L.T. Gardner, “On the Mackey Borel structure”, ''Canad. J. Math.'', '''23''': 4 (1971), pp. 674–678.</TD></TR>
<TR><TD valign="top">[3]</TD><TD valign="top"> H. Halpern, “Mackey Borel structure for the quasi-dual of a separable $ C^{*} $-algebra”, ''Canad. J. Math.'', '''26''': 3 (1974), pp. 621–628.</TD></TR>
</table>

====References====

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<TR><TD valign="top">[a1]</TD><TD valign="top"> W. Arveson, “An invitation to $ C^{*} $-algebras”, Springer (1976), Chapts. 3–4.</TD></TR>
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