# Morita equivalence for *-algebras

This is a reference request. I'm looking for a definition of Morita equivalence of *-algebras, as described below. If anyone thinks that this is not the right way to define Morita equivalence of *-algebras, I'd be interested to hear that also.

By a *-algebra, I mean an algebra equipped with an anti-involution ${}^*$, $(ab)^* = b^*a^*$. (I'm mainly interested in finite-dimensional algebras.)

Recall that a Morita equivalence of algebras $A$ and $B$ consists of (1) a pair of bimodules $_AM_B$ and $_BN_A$, and (2) bimodule isomorphisms $f:{}_A(M\otimes N)_A \to {}_AA_A$ and $g:{}_B(N\otimes M)_B \to {}_BB_B$ such that (3) $f$ and $g$ satisfy the zig-zag identities.

If $A$ and $B$ are *-algebras, it seems to me that the definition of Morita equivalence should be modified (enhanced) as follows.

First, note that the *-structure of $A$ allows us to convert between left and right $A$-modules, and similarly for $B$. In particular, to any $A$-$B$ bimodule $_AX_B$ there is associated a $B$-$A$ bimodule $_BX^*_A$. The first modification to the definition of Morita equivalence is to replace $_BN_A$ above with $_BM^*_A$.

Next, note that both $_AA_A$ and $_A(M\otimes M^*)_A$ have involutions coming from the *-structure. The second (and final) modification to the definition is to require that the isomorphism $f:{}_A(M\otimes M^*)_A \to {}_AA_A$ intertwine with these two involutions, and similarly for $g:{}_B(M^*\otimes M)_B \to {}_BB_B$.

Recall that an (ordinary) Morita equivalence gives an isomorphism between 0-th Hochschild homologies $HH_0(A) \cong HH_0(B)$. A *-structure on $A$ gives rise to an involution of $HH_0(A)$. The isomorphism $HH_0(A) \cong HH_0(B)$ coming from an ordinary Morita equivalence need not commute with the involutions on either side, but the above enhanced Morita equivalences do guarantee compatibility with the involutions on $HH_0$.

So, repeating my initial question, where can i find the above definition in the literature?

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You probably already know this, but a small note: if you are working over the complex numbers, and if your involution is conjugate-linear, there is a slight subtlety to the process of turning left modules into right modules, and vice versa. If $_A M$ is your left module, then the corresponding right module $M^\ast_A$ has to have the complex conjugate vector space $\overline{M}$ as its underlying space. Otherwise the right action of $A$ is conjugate-linear rather than linear. +1 Interesting question! – MTS Dec 25 '12 at 0:53
I'm more interested in the linear (as opposed to conjugate-linear) case, since this is the case that corresponds to 1+1-dimensional unoriented TQFTs. But I'm also interested in the conjugate-linear case you mention, and of course you are right that one has to use the complex conjugate vector space $\overline M$ in that case. – Kevin Walker Dec 25 '12 at 1:56

Perhaps what you want is something along the lines of Rieffel's strong Morita equivalence of $C^\ast$-algebras (see, for instance, http://math.berkeley.edu/~alanw/242papers99/bursztyn, and pretty much any introductory account of operator-algebraic noncommutative geometry), except forgetting all the functional-analytic nuances?

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Interesting idea -- thanks. – Kevin Walker Dec 24 '12 at 18:26

I suggest you look at the book "Morita Equivalence and Continuous-Trace C*-algebras" by I. Raeburn and D. P. Williams. They define much of the construction of a Morita equivalence for -subalgebras of C-algebras without assuming the completeness. You can also look at S. Echterhoff's notes arXiv:1006.4975, where the notion of linking algebra is used to construct Morita equivalence. I have a feeling you can use linking algebras for your purpose as well. Good Luck!

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Thanks for the suggestion. – Kevin Walker Dec 24 '12 at 18:52
Yes, that's true. Of course there is also a notion of Morita equivalence for Banach algebras which is developed mostly by Walter Paravicini. But I do not know whether he considers involutive Banach algebras too. – Vahid Shirbisheh Dec 24 '12 at 18:54