Timeline for Most natural equivalence between $C^*$-algebras in NCG
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 12, 2016 at 0:25 | vote | accept | Melquíades Ochoa | ||
| Apr 7, 2016 at 23:09 | comment | added | truebaran | Could you say something about the notion of equivalence in the sense of Prugovecki? I'm not familiar with this notion. | |
| Apr 5, 2016 at 0:10 | history | edited | truebaran | CC BY-SA 3.0 |
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| Apr 4, 2016 at 20:01 | comment | added | Melquíades Ochoa | @truebaran I see. I get the point that Morita equivalence is a good equivalence for NC-spaces, thanks a lot! Though I would also like to know what do you gain/lose when you consider other equivalences (e.g. Prugovecki)? | |
| Apr 4, 2016 at 19:10 | comment | added | Branimir Ćaćić | One more crucial fact: the crossed product $C_0(X) \rtimes_r G$ of the $C^\ast$-algebra of a locally compact Hausdorff space by a free and proper action of a locally compact topological group is, in general, only Morita equivalent to the $C^\ast$-algebra of the quotient space $C_0(X/G)$. Indeed, $C_0(X) \rtimes_r G$ is necessarily noncommutative the moment $G$ is non-Abelian. | |
| Apr 4, 2016 at 19:04 | comment | added | truebaran | In 2. the answer is negative for $*$-isomorphism just because that if $A$ is commutative then $B$ is not (for $n>1$) so $A$ and $B$ can not be isomorphic. Obviously $*$-isomorphic $C^*$-algebras trivially have same invariants (K-theory, Hochschild homology). In point 4. "same representation theory" means that the category of representations are naturally equivalent (objects are representations and morphism are intertwiners between them). | |
| Apr 4, 2016 at 18:58 | comment | added | Melquíades Ochoa | Do you know negative or positive counterparts of points 2,3,4 for $*$-isomorphism? | |
| Apr 4, 2016 at 18:53 | history | answered | truebaran | CC BY-SA 3.0 |