# 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?

• 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, 2012 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. Dec 25, 2012 at 1:56

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 Crossed products and the Mackey-Rieffel-Green machine (arXiv:1006.4975, also published in "K-Theory for Group C*-Algebras and Semigroup C*-Algebras", Oberwolfach Seminars, vol 47.), 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!

• 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.
– user23860
Dec 24, 2012 at 18:54

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?

Usually I do not want to make to much of advertisement for my own stuff, but here it matches only too well: in a series of papers Henrique Bursztyn and myself developped the theory of Morita equivalence and inner product bimodules for *-algebras in quite some details. This builds on one hand on the work of Ara, already mentioned in Benjamin Steinberg's answer. On the other hand, we were looking for generalizations of Rieffel's notion of strong Morita equivalence since we also wanted to incorporate positivity aspects. Thus our setting was to consider *-algebras over ordered rings like the formal power series used in deformation theory. You can find all this on my homepage together with a (not yet complete) lecture note. Therein you find also a nice categorical framework to compute and describe lots of Morita invariants and a detailed comparison of the various Morita theories (with/out *-involution with/out positivity...).

Have you looked at the following article?

Pere Ara, Morita Equivalence for Rings with Involution, Algebras and Representation Theory 2 (1999) pp 227–247 doi:10.1023/A:1009958527372