### Major advances in the operator amenability problem

#### by Orr Shalit

Laurent Marcoux and Alexey Popov recently published a preprint, whose title speaks for itself :”Abelian, amenable operator algebras are similar to C*-algebras“. This complements another recent contribution, by Yemon Choi, Ilijas Farah and Narutaka Ozawa, “A nonseparable amenable operator algebra which is not isomorphic to a C*-algebra“.

The open problem that these two papers address is whether every amenable Banach algebra, which is a subalgebra of , is similar to a (nuclear) C*-algebra. As the titles clearly indicate (good titling!), we now know that an abelian amenable operator algebra **is** similar to a C*-algebra, and on the other hand, that a non-separable, non-abelian operator algebra **is not necessarily** similar to a C*-algebra.

I recommend reading the introduction to the Marcoux-Popov paper (which is very friendly to non-experts too) to get a picture of this problem, its history, and an outline of the solution.

Those who are interested by the Marcoux-Popov paper may also like to know that James Gifford’s PhD thesis “Operator algebras with a reduction property” is now on the arXiv: http://arxiv.org/abs/1311.3822

Thanks. Indeed, I might have mentioned above that the authors give a lot of credit to ideas from Gifford’s thesis.