View source for Mackey-Borel structure

A Borel structure (i.e. a [[Borel system of sets|Borel system of sets]]) on the spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m0620601.png" /> of a separable [[C*-algebra|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m0620602.png" />-algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m0620603.png" /> (cf. also [[Spectrum of a C*-algebra|Spectrum of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m0620604.png" />-algebra]]), defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m0620605.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m0620606.png" /> be a Hilbert space of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m0620607.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m0620608.png" /> be the set of non-zero irreducible representations (cf. [[Irreducible representation|Irreducible representation]]) of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m0620609.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206010.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206011.png" /> equipped with the topology of pointwise convergence in the weak topology. Let on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206012.png" /> be given the Borel structure generated by its topology (that is, the smallest Borel structure relative to which all mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206016.png" />, are Borel functions) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206017.png" /> be the union of the subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206019.png" /> provided with the Borel structure such that a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206020.png" /> is a Borel set if and only if its intersection with each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206021.png" /> belongs to the Borel structure on the latter. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206022.png" /> be the mapping of the Borel space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206023.png" /> into the spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206024.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206025.png" /> which maps a representation to its unitary equivalence class. The Borel structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206026.png" /> generated by the sets whose inverse images under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206027.png" /> are Borel sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206028.png" /> is called the Mackey–Borel structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206029.png" />. The Mackey–Borel structure contains all sets of the Borel structure generated by the topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206030.png" />; each point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206031.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206032.png" />; 3) the Mackey–Borel structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206033.png" /> is countably separated; and 4) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206034.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206035.png" />-algebra, then a Mackey–Borel structure can also be introduced on the quasi-spectrum of a separable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206036.png" />-algebra.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Dixmier,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206037.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206038.png" />-algebra"  ''Canad. J. Math.'' , '''26''' :  3  (1974)  pp. 621–628</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Arveson,   "An invitation to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062060/m06206039.png" />-algebras" , Springer  (1976)  pp. Chapts. 3–4</TD></TR></table>