Timeline for Correspondences as generalized morphism between $C^*$-algebras
Current License: CC BY-SA 3.0
18 events
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| Feb 2, 2016 at 12:20 | answer | added | user85913 | timeline score: 11 | |
| Jan 11, 2015 at 16:33 | comment | added | Yemon Choi | @BranimirĆaćić +1 just because it warms my heart to see people discussing G-N duality and realizing that maybe one wants to consider continuous maps that aren't proper | |
| Jan 11, 2015 at 16:10 | history | edited | truebaran | CC BY-SA 3.0 |
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| Jan 4, 2015 at 19:20 | comment | added | Branimir Ćaćić | See, for instance, mathoverflow.net/questions/82871/… and similar discussions on MathOverflow and Math.SE. I suppose the point is that naive $\ast$-homomorphisms $A \to B$ aren't necessarily all that natural in the nonunital case, when seen through the guiding lens of Gelfand–Naimark. | |
| Jan 4, 2015 at 19:19 | comment | added | Branimir Ćaćić | I think that the heart of the matter here is that Gelfand–Naimark duality for locally compact Hausdorff spaces is a subtle business. One way to cut the Gordian knot is Woronowicz's approach, which identifies a continuous map $f: X \to Y$ as inducing a nondegenerate $\ast$-homomorphism $f^t : C_0(Y) \to M(C_0(X))$, where $M(C_0(X))$ denotes the multiplier algebra of $C_0(X)$; indeed, if $\phi : A \to M(B)$ is a nondegenerate $\ast$-homomorphism, so that $\phi(A)B$ is dense in $B$, then the induced bimodule ${}_A B_B$ satisfies both nondegeneracy and fullness. | |
| Jan 4, 2015 at 17:10 | comment | added | truebaran | Thank You, this notion as far as I remeber is called fullness. However, while this fixes the problem described in "Edit" but it causes the following new problem: one virtue (which is nice) of this construction is that each ordinary morphism $f:A \to B$ gives rises to the correspondence between $A$ and $B$ as described above. But if we restrict ourselves to modules $X$ with the properties that $AX$ and $\langle X,X \rangle$ are dense in $A$ and $B$ (resp.) I don't see how this ordinary morphism produces the correspondence. | |
| Jan 4, 2015 at 10:00 | comment | added | Branimir Ćaćić | I completely forgot; thanks for reminding me. There is a corresponding condition on the right, which says that $(E,E)$ should be dense in $A$; again, it holds for ${}_A A_A$ precisely because $A$ admits an approximate unit. | |
| Jan 4, 2015 at 0:48 | comment | added | truebaran | Thank you, with such an assumption the argument will follow. I think that it is also necessary to assume that for $_AX_B$ $XB$ is also dense in $X$-while in order to have that $_AA_A$ defines an identity morphism we should also tensor a given correspondence $_BX_A$ from the right. Am I right? | |
| Jan 3, 2015 at 23:41 | comment | added | Branimir Ćaćić | In particular, observe that the “trivial line bundle” ${}_A A_A$ satisfies this condition precisely because $A$ admits an approximate unit. | |
| Jan 3, 2015 at 23:34 | comment | added | Branimir Ćaćić | In the nonunital case, one often imposes the additional condition on ${}_A E_B$ that $A E$ be dense in $E$; this should take care of the issue in your edit. | |
| Jan 3, 2015 at 22:40 | comment | added | truebaran | Dear Alain Valette forgive me, of course I'm only interested in nontrivial correspondences. Dear Dimitri Chikhladze in fact I would like not to go into higher category technicalities. Moreover in this case I am (mainly) interested in nonunital algebras while for unital $C^*$-algebras Rieffel's notion of Morita equivalence coincides with the usual one. | |
| Jan 3, 2015 at 22:25 | comment | added | Dimitri Chikhladze | One needs to consider unital algebras to have a bicategory of bimodules. The nlab page ncatlab.org/nlab/show/Hilbert+bimodule refers to a (2, 1)-category of Hilbert bimodules and their isomorphisms. If you follow the references there you might find an answer to your third question. | |
| Jan 3, 2015 at 21:42 | comment | added | Alain Valette | Ad Q1: Usually homomorphisms are indeed assumed to be *-homomorphisms. Ad Q2: What about taking for $\phi$ the zero homomorphism $A\rightarrow B$? | |
| Jan 3, 2015 at 21:10 | comment | added | truebaran | ...the definition in purely algebraic setting is such that the converse implication: namely that an abstract isomorphism implies Morita equivalence is straightforward (unless in a $C^*$-context). What is more: as I explained, each ordinary morphism $f:A \to B$ gives rise to the associated correspondence. In purely algebraic setup composition of ordinary morphism is compatible with composition (tensor product) of bimodules: however in the $C^*$-algebra situation I met exactly the same problem (which I described after "EDIT") while trying to prove such a compatibility. | |
| Jan 3, 2015 at 21:03 | comment | added | truebaran | To be honest, I'm a little afraid of higher category theory. But let me clarify: after getting some familiarity with the notion of Morita equivalence in purely algebraic setting I learned that one can speak about the category $\tilde{\textbf{Alg}}$ where objects are unital algebras and morphisms between $A$ and $B$ are isomorphic classes of $A-B$ bimodules. Then isomorphism in this abstract category is exactly Morita equivalence. Composition is of course defined by tensor product: while we deal with unital algebras one can easily check that $_AA_A$ defines the identity morphism. Moreover... | |
| Jan 3, 2015 at 20:44 | comment | added | Dimitri Chikhladze | Btw, it seems that you are talking about a bicategory of correspondences, rather than just a category. | |
| Jan 3, 2015 at 19:45 | history | edited | truebaran | CC BY-SA 3.0 |
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| Jan 3, 2015 at 1:48 | history | asked | truebaran | CC BY-SA 3.0 |