Mackey-Borel structure

A Borel structure (i.e. a Borel system of sets) on the spectrum of a separable -algebra (cf. also Spectrum of a -algebra), defined as follows. Let , be a Hilbert space of dimension , and let be the set of non-zero irreducible representations (cf. Irreducible representation) of the -algebra on equipped with the topology of pointwise convergence in the weak topology. Let on be given the Borel structure generated by its topology (that is, the smallest Borel structure relative to which all mappings , , , , are Borel functions) and let be the union of the subspaces , provided with the Borel structure such that a subset of is a Borel set if and only if its intersection with each belongs to the Borel structure on the latter. Let be the mapping of the Borel space into the spectrum of which maps a representation to its unitary equivalence class. The Borel structure on generated by the sets whose inverse images under are Borel sets in is called the Mackey–Borel structure on . The Mackey–Borel structure contains all sets of the Borel structure generated by the topology of ; each point of is a Borel set in the Mackey–Borel structure. The following 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 ; 3) the Mackey–Borel structure on is countably separated; and 4) if is a -algebra, then a Mackey–Borel structure can also be introduced on the quasi-spectrum of a separable -algebra.

References

[1]	J. Dixmier, " 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 -algebra" Canad. J. Math. , 26 : 3 (1974) pp. 621–628

References

[a1]	W. Arveson, "An invitation to -algebras" , Springer (1976) pp. Chapts. 3–4