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
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
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
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
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
--- 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. Ask a question on any topic and get answers from real people. Go to Yahoo! Answers and share what you know at http://ca.answers.yahoo.com
participants (6)
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Bas Spitters -
Gabor Lukacs -
Jeff Egger -
John Baez -
Miles Gould -
Yemon Choi