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.
1-apr-2006
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
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/29940
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 4
  • ???jsp.display-item.citation.isi??? 2
social impact