Strong Morita equivalence and representation theory

Question

In the context of pure algebra we say that two algebras (in general: rings) $A,B$ are Morita equivalence when there are bimodules $_AP_B,_BQ_A$ such that $P \otimes_B Q \cong _AA_A$ and $Q \otimes_A P \cong _BB_B$. The remarkable theorem states that this condition is equivalent to the following fact: the categories of (for example) left modules over $A$ and over $B$ are naturally equivalent. There is a notion of so called strong Morita equivalence due to Rieffel: this notion is for the $C^*$-algebra context. I saw the following remark: two Morita equivalent $C^*$-algebras have the same representation theory. I wonder what exactly does it mean: I'm interested in $*$-representations. In particular, is it true that $*$-representations of the $C^*$-algebra somehow form the category and the notion of strong Morita equivalence is the natural equivalence of these categories?

Answer 1

Let $A$ and $B$ be two $C^{*}$ algebras. Then the category of $*$-representation of $A$ on Hilbert space is equivalent to that of $B$ if and only if their enveloping von Neumann algebra are morita equivalent (note that the notion of morita equivalence is slightly more restrictive for Von Neumann algebras because we expect the bi-modules to be self dual as module).

In particular, two commutative algebras with the same enveloping von Neuman algebras are going to have the same representations categories but are not Morita equivalent (unless they are isomorphic of course)

The good notion of module to have a similar result for $C^*$-algebra are the Hilbert Modules : two $C^*$-algebra are Morita equivalent if and only if they have equivalent categories of Hilbert modules (equivalent as C* categories).

Oh, but the converse holds: two Morita equivalent $C^*$ algebras do have the same category of representations (for exemple, because the equivalence bi-module between them can be completed into a self-dual equivalence bi-module between their enveloping algebras)

Some references:

The first references are I think the two papers of Rieffel:

Morita equivalence for C* algebras and W* algebras

Morita equivalence for operator algebras

In the first paper (and contrary to what the title suggest) he deals with morita equivalence for von Neumann algebras (what he calls morita equivalence of C* algebra in this paper corresponds to the weak notion and hence to morita equivalences of the enveloping W*-algebras) In the second paper he talks about strong morita equivalence of C* algebras. He didn't explicitly proves that it the same as the equivalence of the categories of Hilbert modules but He makes some remarks that essentially explain how it works.

Bruce Blackadar's "Operator Algebras" give a nice and modern acount of the basic theory of Morita equivalence (section II.7) but didn't seems to prove that it is equivalent to the equivalence of the categories of C* module neither...

Unfortunately, I haven't been able to find an actual complete and explicit proof in the literature of this last results. (I will edit again if I found one)