--- 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