A Morita type equivalence for dual operator algebras

Abstract

We generalize the main theorem of Rieffel for Morita equivalence of W * -algebras to the case of unital dual operator algebras: two unital dual operator algebras A, B have completely isometric normal representations α, β such that α(A) = [M * β(B)M] −w * and β(B) = [Mα(A)M * ] −w * for a ternary ring of operators M (i.e. a linear space M such that MM * M ⊂ M) if and only if there exists an equivalence functor F : A M → B M which "extends" to a * -functor implementing an equivalence between the categories A DM and B DM. By A M we denote the category of normal representations of A and by A DM the category with the same objects as A M and ∆(A)-module maps as morphisms (∆(A) = A ∩ A * ). We prove that this functor is equivalent to a functor "generated" by a B, A bimodule, and that it is normal and completely isometric.

Stably isomorphic dual operator algebras

“…We prove that two unital dual operator algebras A, B are stably isomorphic if and only if they are ∆-equivalent [7], if and only if they have completely isometric normal representations α, β on Hilbert spaces H, K respectively and there exists a ternary ring of operators M ⊂ B(H, K) such that α(A) = [M * β(B)M] …”

“…In [7] the notion of ∆-equivalence of two unital dual operator algebras A, B was defined in terms of equivalence of two appropriate categories. In the present paper, we will adopt the following definition of ∆-equivalence.…”

Morita equivalence of dual operator algebras

“…Carefully computing the first, and then the second, commutants of L w (as in 8.5.32 in [8] or [22]), and using the double commutant theorem, gives the result. coincides also with the equivalence of [17,18,20], that is, weak* stable isomorphism. Indeed if (M, N, X, Y) is a weak* Morita context, then it is clearly a strong Morita context in this case.…”

“…Then the weak linking algebra of such an example is Morita equivalent in the same sense to A (see Section 4), but they are probably not always weak* stably isomorphic. (10) A beautiful example from [19] (formerly part of [17]): two 'similar' separably acting nest algebras are clearly weakly Morita equivalent by the facts presented around [19, Theorem 3.5] (Davidson's similarity theorem), indeed in this case the 'subcontext' (see Definition 3.2) equals the 'context', and the algebras are even strongly Morita equivalent. However, Eleftherakis shows they need not be '∆-equivalent' (that is, weak* stably isomorphic [20]).…”

“…We thank Vern Paulsen for suggesting that we look again at the last example in [19] (which was previously part of [17]). Indeed this is clearly a weak (indeed even a strong) Morita equivalence, by Davidson's similarity theorem (see Example (10) in Section 3).…”