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Haagerup approximation property for arbitrary von Neumann algebras. (English) Zbl 1335.46052
A finite von Neumann algebra \(M\) is said to have the Haagerup approximation property if for some (equivalently, every) faithful normal tracial state \(\tau\) on \(M\) there exists a net of unital completely positive \(\tau\)-preserving maps \(\Phi_i\) converging to the identity in the point weak\(^*\)-topology and such that their natural \(L^2(M, \tau)\) versions are compact. This notion, motivated by the Haagerup property for discrete groups, was introduced by M. Choda [Proc. Japan Acad., Ser. A 59, 174–177 (1983; Zbl 0523.46038)] and further studied by P. Jolissaint [J. Oper. Theory 48, No. 3, 549–571 (2002; Zbl 1029.46091)].
In the present article, the authors extend this notion to arbitrary \(\sigma\)-finite von Neumann algebras, with the standard form framework as the starting point – so that the defining approximations are first viewed as operators \(T_i\) on the canonical Hilbert space on which the von Neumann algebra acts. It is shown that in the finite case the notion coincides with that of M. Choda [Proc. Japan Acad., Ser. A 59, 174–177 (1983; Zbl 0523.46038)] and P. Jolissaint [J. Oper. Theory 48, No. 3, 549–571 (2002; Zbl 1029.46091)] and that given a faithful normal state \(\phi\) one can always extract out of \(T_i\) a corresponding family \(\Phi_i\) of completely positive contractive \(\phi\)-preserving maps on the von Neumann algebra. The property is shown to be stable under taking crossed products by locally compact abelian groups, and under passing to expected subalgebras (even if the corresponding conditional expectation is not normal).
For further information on the recent developments on the Haagerup property for arbitrary von Neumann algebras we refer to [M. Caspers and A. Skalski, Commun. Math. Phys. 336, No. 3, 1637–1664 (2015; Zbl 1330.46057); Int. Math. Res. Not. 2015, No. 19, 9857–9887 (2015; Zbl 1344.46043)], another forthcoming preprint by the authors, entitled “Haagerup approximation property and positive cones associated with a von Neumann algebra”, arXiv:1403.3971, and their joint preprint with Narutaka Ozawa, “Haagerup approximation property via bimodules”, arXiv:1501.06293.

MSC:
46L10 General theory of von Neumann algebras
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