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Sparked by Yemon Choi's answer to Is the category of Banach spaces with contractions an algebraic theory? I've just spent a merry time reading and doing a bit of reference chasing. Imagine my delight at finding that one of my old favourites (functional analysis) and one of my new fads (category theory, and in particular algebraic theories) are actually very closely connected!

I was going to ask about the state of play of these things as it's a little unclear exactly what stage has been achieved. Reading the paper On the equational theory of $C^\ast$-algebras and its review on MathSciNet then it appears that although it's known that $C^\ast$-algebras do form an algebraic theory, an exact presentation in terms of operations and identities is still missing (at least at the time of that paper being written), though I may be misreading things there. It's possible to do a little reference chasing through the MathSciNet database, but the trail does seem to go a little cold and it's very hard to search for "$C^\ast$ algebra"!

But now I've decided that I don't want to just know about the current state of play, I'd like to learn what's going on here in a lot more detail since, as I said, it brings together two seemingly disparate areas of mathematics both of which I quite like.

So my real question is

  • Where should I start reading?

Obviously, the paper Yemon pointed me to is one place to start but there may be a good summary out there that I wouldn't reach (in finite) time by a reference chase starting with that paper. So, any other suggestions? I'm reasonably well acquainted with algebraic theories in general so I'm looking for specifics to this particular instance.

Also, I'll write up my findings as I find them on the n-lab so anyone who wants to join me is welcome to follow along there. I probably won't actually start until the new year though.

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Do an emath search for Waelbroeck, L*; note especially his paper "The Taylor spectrum and quotient Banach spaces". For more recent things, search for Castillo, J*. Also, Mariusz Wodzicki at Berkeley has unpublished notes that contain many things. I don't know if they are in a form for distribution.

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