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On commutative, operator amenable subalgebras of finite von Neumann algebras. (English) Zbl 1282.46054
The paper under review deals with a problem whose origins and motivations are classic results of Dixmier, Day and others dating from 1950. Let us recall that a Banach algebra \(A\) is called amenable when every bounded derivation from \(A\) into a dual Banach \(A\)-bimodule is inner (i.e., it is of the form \(a\mapsto a.x-x.a\) for some \(x\) in the dual module). This definition is due to B. E. Johnson, who also established in a subsequent paper that \(A\) is amenable if and only if it has a virtual diagonal [Am. J. Math. 94, 685–698 (1972; Zbl 0246.46040)]. The contributions of A. Connes [J. Funct. Anal. 28, 248–253 (1978; Zbl 0408.46042)] and U. Haagerup [Invent. Math. 74, 305–319 (1983; Zbl 0529.46041)] show that a \(C^*\)-algebra is amenable if and only if it is nuclear.
It is clear that the definition of amenability does not make any use of any involution, however, in contrast to the \(C^*\)-setting, up to date there is no comparable characterization of amenable, not necessarily self-adjoint operator algebras. A method to produce examples of amenable Banach algebras is described as follows. Let \(A\) be an amenable \(C^*\)-algebra faithfully represented on a Hilbert space \(H\) and let \(R\) be an invertible element in \(L(H)\), then the algebra \(R^{-1} A R\) is an amenable algebra. It has been asked by several authors, in various forms and in various contexts, if every amenable operator algebra arises in this way. The conjecture remains open in the general case, but there are notable partial results by J. A. Gifford [J. Aust. Math. Soc. 80, No. 3, 297–315 (2006; Zbl 1103.46031)], G. A. Willis [J. Oper. Theory 34, No. 2, 239–249 (1995; Zbl 0855.46029)], L. W. Marcoux [J. Lond. Math. Soc., II. Ser. 77, No. 1, 164–182 (2008; Zbl 1145.46032)], and P. C. Curtis jun. and R. J. Loy [Bull. Aust. Math. Soc. 52, No. 2, 327–329 (1995; Zbl 0836.47034)]. This paper presents new evidence to support this conjecture, showing that a closed, commutative, operator amenable subalgebra of a finite von Neumann algebra is similar to a self-adjoint subalgebra. Since an amenable Banach algebra is operator amenable, the author also shows that a closed, commutative, amenable subalgebra of a finite von Neumann algebra is similar to a self-adjoint subalgebra, providing a positive answer to the above conjecture in this particular setting.

46L10 General theory of von Neumann algebras
46L06 Tensor products of \(C^*\)-algebras
46L07 Operator spaces and completely bounded maps
46L08 \(C^*\)-modules
46L57 Derivations, dissipations and positive semigroups in \(C^*\)-algebras
47L40 Limit algebras, subalgebras of \(C^*\)-algebras
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