From: John Baez <baez@math.ucr.edu>
To: categories <categories@mta.ca>
Subject: Re: C*-algebras
Date: Mon, 30 Apr 2007 13:54:25 -0700 [thread overview]
Message-ID: <E1HiqUm-0000kU-Sb@mailserv.mta.ca> (raw)
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
next reply other threads:[~2007-04-30 20:54 UTC|newest]
Thread overview: 6+ messages / expand[flat|nested] mbox.gz Atom feed top
2007-04-30 20:54 John Baez [this message]
-- strict thread matches above, loose matches on Subject: below --
2007-05-02 17:03 C*-algebras Jeff Egger
2007-05-01 15:34 C*-algebras Miles Gould
2007-04-29 4:30 C*-algebras Gabor Lukacs
2007-04-29 3:26 C*-algebras Yemon Choi
2007-04-28 20:27 C*-algebras Bas Spitters
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