# Isomorphism of categories of rigged modules via completely bounded isomorphism of operator algebras

This question is a background for my previous question.

Suppose $A$ and $B$ are two algebras over $\mathbb{C}$ with the sequences of norms $\lbrace\|\cdot\|_{\Xi,n}\rbrace$ and on $M_n(\Xi)$, $\Xi\in\lbrace A, B\rbrace$, satisfying the conditions of Blecher-Ruan-Sinclar theorem (so that, if I understand it right, we may construct concrete representations). Suppose also that $f\colon A \to B$ is a completely bounded map that has a completely bounded inverse $f^{-1}\colon B\to A$.

Can we somehow establish an isomorphism between categories of rigged modules over $A$ and $B$. And if yes, is there any good reference?

It can probably fit into the notion of (P)-context, but I can't reach the book right now to check all the conditions.

• The theorem of Ruan to which you link is for completely isometrically representing operator spaces. The corresponding result for operator algebras came a little later from Blecher, Ruan, and Sinclair. See e.g. jstor.org/pss/2160382. – Jonas Meyer Aug 11 '10 at 22:25
• So, a good question is: what extra conditions on the matrix norms do you want to impose? You could insist that A and B become completely bounded (or contractive) Banach algebras. Or, a strong condition is that M_n(A) becomes a contractive algebra for each n: then the work which Jonas eludes to shows that A must be a (non-self-adjoint) operator algebra. – Matthew Daws Aug 12 '10 at 9:02
• To Matthew Daws. Well, I supposed to have the second, stronger condition, with the possibility for A to become an operator algebra. – Kolya Ivankov Aug 12 '10 at 9:22