Re: C*-algebras

Re: C*-algebras

[Gabor Lukacs]

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

Re: C*-algebras

[Jeff Egger]

--- Miles Gould <miles@assyrian.org.uk> wrote:

> Jeff Egger gave a talk on some of these ideas at the Nice PSSL - I'm
> somewhat surprised he hasn't replied to this thread.

I'm usually too shy to post to the mailing list, so I wrote Bas Spitters
a personal reply instead.  In this case, though, I have to set the
record straight...

> IIRC, the category
> of operator algebras is an involutive monoidal category with respect to
> one or other of the tensor products, and C*-algebras are exactly the
> involutive monoids w.r.t. this tensor product. Can't remember which one
> it was, though.

The category of operator _spaces_ admits a (non-trivial) involutive monoidal
structure---by which I mean a (non-commutative) monoidal structure together
with a _covariant_ involution that reverses the order of tensoring.
[Regarding a monoidal category as a one-object bicategory B, this means
that the involution relates B with B^{op} rather than with B^{co}.]
The tensor product is called the _Haagerup_ tensor product, and the
involution I considered is the so-called _opposite_ operator space
structure applied to the conjugate vector space.  I had conjectured that
involutive monoids in this involutive monoidal category (which, for the
purposes of this mail, I shall call involutive operator algebras) are
the same as C*-algebras, but eventually I discovered a counter-example
which showed that involutive operator algebras are strictly more general
than C*-algebras.  (This was the direction which had less concerned me!)
I apologise to anyone to whom I failed to mention this counter-example.

Cheers,
Jeff.



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Re: C*-algebras

[Miles Gould]

On Sat, Apr 28, 2007 at 10:27:58PM +0200, 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. Which one is `the right one' from a categorical perspective?

Jeff Egger gave a talk on some of these ideas at the Nice PSSL - I'm
somewhat surprised he hasn't replied to this thread. IIRC, the category
of operator algebras is an involutive monoidal category with respect to
one or other of the tensor products, and C*-algebras are exactly the
involutive monoids w.r.t. this tensor product. Can't remember which one
it was, though.

Miles

Re: C*-algebras

[John Baez]

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

Re: C*-algebras

[Yemon Choi]

I'm not an expert but I don't think there is a `right one', it depends
on what you want to do with your C*-algebras. The maximal C*-norm has
better universal properties than the minimal one (it seems) but the
resulting C*-algebra is then somewhat hard to get at.

Actually, I'm not sure what significance the tensor product of
_algebras_ (as opposed to _modules_) has. Of course for commutative
unital algebras this is the coproduct, but commutative C*-algebras
have unique C*-tensor norms anyway.



On 28/04/07, Bas Spitters <B.Spitters@cs.ru.nl> wrote:
> It seems hard to find references to a categorical treatment of
> C*-algebras. Concretely, there are several tensor products on
> C*-algebras. Which one is `the right one' from a categorical perspective?
>
> Thanks,
>
> Bas Spitters
>
>


-- 
Dr. Y. Choi
519 Machray Hall
Department of Mathematics
University of Manitoba
Winnipeg. Manitoba
Canada R3T 2N2

C*-algebras

[Bas Spitters]

It seems hard to find references to a categorical treatment of
C*-algebras. Concretely, there are several tensor products on
C*-algebras. Which one is `the right one' from a categorical perspective?

Thanks,

Bas Spitters