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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.
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A Borel structure (i.e., a [[Borel system of sets|Borel system of subsets]]) on the spectrum 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:
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# 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).
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# The Mackey–Borel structure coincides with the Borel structure generated by the topology on \widehat{A} .
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# The Mackey–Borel structure on \widehat{A} is countably separated.
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# 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====
 
====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>
 
  
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<table>
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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>
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<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>
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<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>
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</table>
  
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====References====
  
====Comments====
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<table>
 
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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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</table>
====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>
 

Latest revision as of 04:32, 4 December 2016

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.
How to Cite This Entry:
Mackey-Borel structure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mackey-Borel_structure&oldid=18109
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article