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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 $ \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:
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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=22785
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article