## On commutative, operator amenable subalgebras of finite von Neumann algebras

#### Authors

Yemon Choi

Keywords:

MSC 2010: primary 46J40, 47L75; secondary 47L60.

DOI: 10.1515/crelle.2012.030

#### Status

Appeared as J. Reine. Angew. Math. 678 (2013), 201--222

Preprint version available at arXiv 1012.4259. Small differences from the final accepted version, including some differences in numbering of theorems etc.

#### Abstract

It has been conjectured, motivated in part by old results of Dixmier and Day on bounded Hilbertian representations of amenable groups, that every norm-closed amenable subalgebra of ${\mathcal B}({\mathcal H})$ is automatically similar to an amenable C*-algebra. Results of Curtis and Loy (1995), Gifford (2006), and Marcoux (2008) give some evidence to support this conjecture, but it remains open even for commutative subalgebras.

We present more evidence to support this conjecture, by showing that a closed, commutative, operator amenable subalgebra of a finite von Neumann algebra ${\mathcal M}$ must be similar to a selfadjoint subalgebra. Technical results used include an approximation argument based on Grothendieck's inequality and the Pietsch domination theorem, together with an adaptation of a theorem of Gifford (ibid.) to the setting of unbounded operators affiliated to ${\mathcal M}$.

That is: they proved that every amenable commutative (norm-closed) subalgebra of B(H) is similar to a self-adjoint one. It is worth noting that like my partial result, all the hard work goes into proving that the subspace norm on the algebra A is equivalent to the spectral radius in A; this implies that the Gelfand representation $A \to C_0(\Phi_A)$ is bounded below, and from there one appeals to Sheinberg's theorem and Kadison similarity, just as is in my paper. However, the approach found by Marcoux and Popov is simpler than mine, even when one assumes A is contained in a finite von Neumann algebra (their arguments are purely global and do not require any type decomposition).