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From: John Baez <baez@math.ucr.edu>
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Subject: Re: C*-algebras
Date: Mon, 30 Apr 2007 13:54:25 -0700
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Bas Spitters wrote:

>It seems hard to find references to a categorical treatment of
>C*-algebras. Concretely, there are several tensor products on
>C*-algebras.

The two I know are the "projective" and "injective" tensor products.

> Which one is 'the right one' from a categorical perspective?

I think the projective (or "maximum possible norm") tensor product
of unital C*-algebras has the following universal property:

There are homomorphisms from A and B into the projective tensor
product A tensor B, and given homomorphisms f: A -> X,
g: B -> X whose ranges commute, there exists a unique homomorphism

f tensor g: A tensor B -> X

such that the two obvious triangles commute, namely one like this:

A ------> A tensor B
  \         |
   \        |
    \       |
    f\      |f tensor g
      \     |
       \    |
        v   v
          X

and a similar one for B.

Here I'm using the unital nature of the C*-algebras in question
to get the homomorphisms from A and B into A tensor B; you have
to do something different for nonunital C*-algebras.

Best,
jb