Re: C*-algebras

[Gabor Lukacs <lukacs@cc.umanitoba.ca>]

Dear Bas,


> It seems hard to find references to a categorical treatment of
> C*-algebras.

I am surprised to hear that. Here are a few, which I am almost sure that 
you are already familiar with:

D. H. Van Osdol. C*-algebras and cohomology. In Categorical topology
(Toledo, Ohio, 1983), volume 5 of Sigma Ser. Pure Math., pages 582-587.
Heldermann, Berlin, 1984.

J. Wick Pelletier and J. Rosick´y. On the equational theory of 
C*-algebras. Algebra Universalis, 30(2):275-284, 1993.

Joan Wick Pelletier and Ji.r´. Rosick´y. Generating the equational theory 
of C*-algebras and related categories. In Categorical topology and its 
relation to analysis, algebra and combinatorics (Prague, 1988), pages 
163-180. World Sci. Publishing, Teaneck, NJ, 1989.

Edward G. Effros and Zhong-Jin Ruan. Operator spaces, volume 23 of
London Mathematical Society Monographs. New Series. The Clarendon
Press Oxford University Press, New York, 2000.

[This last one is not categorical, but it contains some results concerning 
the tensor products that can easily be interpreted categorically.]

You my find a brief summary of the categorically interesting points in 
Chapter 8 of my PhD thesis:

http://at.yorku.ca/p/a/a/o/41.pdf

> Concretely, there are several tensor products on C*-algebras. Which one 
> is `the right one' from a categorical perspective?

This is an interesting question, but I suspect that you may find a clue to 
answer this question here:

Theodore W. Palmer. Banach algebras and the general theory of *-algebras. 
Vol. 2, volume 79 of Encyclopedia of Mathematics and its Applications.
Cambridge University Press, Cambridge, 2001.

I hope that my answers are of some help to you.

Best wishes,

Gabi