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Register a new userStable isomorphism and strong Morita equivalence of operator algebras
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Authors
G. K. Eleftherakis
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We introduce a Morita type equivalence: two operator algebras $A$ and $B$ are called strongly $Delta $-equivalent if they have completely isometric representations $alpha $ and $beta $ respectively and there exists a ternary ring of operators $M$ such that $alpha (A)$ (resp. $beta (B)$) is equal to the norm closure of the linear span of the set $M^*beta (B)M, $ (resp. $Malpha (A)M^*$). We study the properties of this equivalence. We prove that if two operator algebras $A$ and $B,$ possessing countable approximate identities, are strongly $Delta $-equivalent, then the operator algebras $Aotimes cl K$ and $Botimes cl K$ are isomorphic. Here $cl K$ is the set of compact operators on an infinite dimensional separable Hilbert space and $otimes $ is the spatial tensor product. Conversely, if $Aotimes cl K$ and $Botimes cl K$ are isomorphic and $A, B$ possess contractive approximate identities then $A$ and $B$ are strongly $Delta $-equivalent.
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We define a relation < for dual operator algebras. We say that B < A if there exists a projection p in A such that B and pAp are Morita equivalent in our sense. We show that < is transitive, and we investigate the following question: If A < B and B < A, then is it true that A and B are stably isomorphic? We propose an analogous relation < for dual operator spaces, and we present some properties of < in this case.
We define a strong Morita-type equivalence $sim _{sigma Delta }$ for operator algebras. We prove that $Asim _{sigma Delta }B$ if and only if $A$ and $B$ are stably isomorphic. We also define a relation $subset _{sigma Delta }$ for operator algebras. We prove that if $A$ and $B$ are $C^*$-algebras, then $Asubset _{sigma Delta } B$ if and only if there exists an onto $*$-homomorphism $theta :Botimes mathcal K rightarrow Aotimes mathcal K,$ where $mathcal K$ is the set of compact operators acting on an infinite dimensional separable Hilbert space. Furthermore, we prove that if $A$ and $B$ are $C^*$-algebras such that $Asubset _{sigma Delta } B$ and $Bsubset _{sigma Delta } A $, then there exist projections $r, hat r$ in the centers of $A^{**}$ and $B^{**}$, respectively, such that $Arsim _{sigma Delta }Bhat r$ and $A (id_{A^{**}}-r) sim _{sigma Delta }B(id_{B^{**}}-hat r). $
We define an equivalence relation between bimodules over maximal abelian selfadjoint algebras (masa bimodules) which we call spatial Morita equivalence. We prove that two reflexive masa bimodules are spatially Morita equivalent iff their (essential) bilattices are isomorphic. We also prove that if S^1, S^2 are bilattices which correspond to reflexive masa bimodules U_1, U_2 and f: S^1rightarrow S^2 is an onto bilattice homomorphism, then: (i) If U_1 is synthetic, then U_2 is synthetic. (ii) If U_2 contains a nonzero compact (or a finite or a rank 1) operator, then U_1 also contains a nonzero compact (or a finite or a rank 1) operator.
We review Morita equivalence for finite type $k$-algebras $A$ and also a weakening of Morita equivalence which we call stratified equivalence. The spectrum of $A$ is the set of equivalence classes of irreducible $A$-modules. For any finite type $k$-algebra $A$, the spectrum of $A$ is in bijection with the set of primitive ideals of $A$. The stratified equivalence relation preserves the spectrum of $A$ and also preserves the periodic cyclic homology of $A$. However, the stratified equivalence relation permits a tearing apart of strata in the primitive ideal space which is not allowed by Morita equivalence. A key example illustrating the distinction between Morita equivalence and stratified equivalence is provided by affine Hecke algebras associated to extended affine Weyl groups.
We prove that for every group $G$ and any two sets $I,J$, the Brandt semigroup algebras $ell(B(I,G))$ and $ell(B(J,G))$ are Morita equivalent with respect to the Morita theory of self-induced Banach algebras introduced by Gronbaek. As applications, we show that if $G$ is an amenable group, then for a wide class of Banach $ell(B(I,G))$-bimodules $E$, and every $n>0$, the bounded Hochschild cohomology groups $H^n(ell(B(I,G)),E^*)$ are trivial, and also, the notion of approximate amenability is not Morita invariant.
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