Yemon Choi

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

DOI: 10.1515/crelle.2012.030

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.

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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 ${\mathcal M}$.

The work in this paper was originally motivated by the 1995 paper of Willis and some subsequent work of Marcoux. Discussions with Matthew Daws in 2006 were also helpful, though at the time we were still thinking about the case of an operator algebra generated by a single operator on Hilbert space. The idea of focusing on finite von Neumann algebras occurred to me some years later; seeking distraction from other research, I found myself re-reading Brown's original article on spectral measure for operators in semifinite von Neumann algebras, and realized that one could mimic Willis's use of Lidskii in the trace-class setting to get something analogous in the II_{1} setting.

The paper was submitted in 2010 and accepted in 2011. A few years later, Marcoux and Popov found a beautiful resolution of the amenable operator algebra problem in the commutative case:

L. W. Marcoux, A. I. Popov, Abelian, amenable operator algebras are similar to C*-algebras. Duke Math. J. 165 (2016), no. 12, 2391--2406; arXiv 1311.2982

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).

Yemon Choi Last updated: 2020-05-25