Let $A$ and $B$ be $C^*$-algebras, I had a question about Morita equivalence for $C^*$-algebras and $W^*$-algebras, I've come across 3 concepts:
- Morita equivalence for $C^*$-algebras: Equivalence of categories of Hermitian modules of $A$ and $B$.
- Strong Morita equivalence for $C^*$-algebras: This is given by the existence of an "$A$-$B$-imprimitivity bimodule" ($A$-$B$-bimodule satisfying certaing conditions).
- Morita equivalence for $W^*$-algebras (as defined by Rieffel in "Morita equivalence for $C^*$-algebras and $W^*$-algebras", this in the case $A$ and $B$ are $W^*$-algebras): This is given by the existence of an "$A$-$B$-equivalence bimodule".
$W^*$-algebras are $C^*$-algebras so what's the difference there? I read somewhere that two $C^*$-algebras $A$ and $B$ are strong Morita equivalent if they are Morita equivalent in the traditional algebraic sense (as rings), but I can't find the proof, so I was wondering:
Are two $W^*$-algebras $A$ and $B$ that are Morita equivalent (due to the existence of an equivalence bimodule), also algebraically Morita equivalent (as rings)?
Anyone have references for this? Are equivalence bimodules and imprimitivity bimodules the same thing? I've seen different definitions of them, some very similar.