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