It is proved that the family of equivalence classes of Lip-normed C*-algebras introduced by M. Rieffel, up to isomorphisms preserving the Lip-seminorm, is not complete w.r.t. the matricial quantum Gromov-Hausdorff distance introduced by D. Kerr. This is shown by exhibiting a Cauchy sequence whose limit, which always exists as an operator system, is not completely order isomorphic to any C*-algebra. Conditions ensuring the existence of a C*-structure on the limit are considered, making use of the notion of ultraproduct. More precisely, a necessary and sufficient condition is given for the existence, on the limiting operator system, of a C*-product structure inherited from the approximating C*-algebra. Such condition can be considered as a generalisation of the f-Leibniz conditions introduced by Kerr and Li. Furthermore, it is shown that our condition is not necessary for the existence of a C*-structure tout court, namely there are cases in which the limit is a C*-algebra, but the C*-structure is not inherited.

Guido, D., & Isola, T. (2006). The problem of completeness for Gromov-Hausdorff metrics on C*-algebras. JOURNAL OF FUNCTIONAL ANALYSIS, 233(1), 173-205 [10.1016/j.jfa.2005.04.007].

The problem of completeness for Gromov-Hausdorff metrics on C*-algebras

GUIDO, DANIELE;ISOLA, TOMMASO
2006-04-01

Abstract

It is proved that the family of equivalence classes of Lip-normed C*-algebras introduced by M. Rieffel, up to isomorphisms preserving the Lip-seminorm, is not complete w.r.t. the matricial quantum Gromov-Hausdorff distance introduced by D. Kerr. This is shown by exhibiting a Cauchy sequence whose limit, which always exists as an operator system, is not completely order isomorphic to any C*-algebra. Conditions ensuring the existence of a C*-structure on the limit are considered, making use of the notion of ultraproduct. More precisely, a necessary and sufficient condition is given for the existence, on the limiting operator system, of a C*-product structure inherited from the approximating C*-algebra. Such condition can be considered as a generalisation of the f-Leibniz conditions introduced by Kerr and Li. Furthermore, it is shown that our condition is not necessary for the existence of a C*-structure tout court, namely there are cases in which the limit is a C*-algebra, but the C*-structure is not inherited.
Pubblicato
Rilevanza internazionale
Articolo
Sì, ma tipo non specificato
Settore MAT/05 - Analisi Matematica
English
Con Impact Factor ISI
quantum metric spaces; ultraproducts; lip-norms
Guido, D., & Isola, T. (2006). The problem of completeness for Gromov-Hausdorff metrics on C*-algebras. JOURNAL OF FUNCTIONAL ANALYSIS, 233(1), 173-205 [10.1016/j.jfa.2005.04.007].
Guido, D; Isola, T
Articolo su rivista
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2108/29940
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