The isomorphism problem: update

Ken Davidson, Chris Ramsey and I recently uploaded a new version of our paper “Operator algebras for analytic varieties” to the arxiv. This is the second paper that was affected by a discovery of a mistake in the literature, which I told about in the previous post. Luckily, we were able to save all the results in that paper, but had to work a a little harder than what we thought was needed in our earlier version. The isomorphism problem for complete Pick algebras (which I like to call simply “the isomorphism problem”) has been one of my favorite problems during the last five years. I wrote four papers on this problem, with five co-authors. I want to give a short road-map to my work on this problem. Before I do so, here is  link to the talk that I will give in IWOTA 2014 about this stuff. I think (hope) this talk is a good introduction to the subject. The problem is about the classification of a large class of non-selfadjoint operator algebras – multiplier algebras of complete Pick spaces – which can also be realized as certain algebras of functions on analytic varieties. These algebras all have the form

M_V = Mult(H^2_d)\big|_V

where V is a subvariety of the unit ball and Mult(H^2_d)  denotes the multiplier algebra of Drury-Arveson space (see this survey), and therefore M_V is the space of all restrictions of multipliers to V. The hope is to show that the geometry of the variety V is a complete invariant for the algebras M_V, in various senses that will be made precise below.

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