Factorization of completely bounded bilinear operators and injectivity

We characterize injectivity of von Neumann algebras in terms of factoring bilinear maps as products of linear maps.Comment: 32 pages. See also http://www.math.tamu.edu/~roger.smith/ preprints.htm

Strong singularity for subalgebras of finite factors

In this paper we develop the theory of strongly singular subalgebras of von Neumann algebras, begun in earlier work. We mainly examine the situation of type \tto factors arising from countable discrete groups. We give simple criteria for strong singularity, and use them to construct strongly singular subalgebras. We particularly focus on groups which act on geometric objects, where the underlying geometry leads to strong singularity

Representations of completely bounded multilinear operators

AbstractA definition of a completely bounded multilinear operator from one C∗-algebra into another is introduced. Each completely bounded multilinear operator from a C∗-algebra into the algebra of bounded linear operators on a Hilbert space is shown to be representable in terms of ∗-representations of the C∗-algebra and interlacing operators. This result extends Wittstock's Theorem that decomposes a completely bounded linear operator from a C∗-algebra into an injective C∗-algebra into completely positive linear operators

Kadison-Kastler stable factors

A conjecture of Kadison and Kastler from 1972 asks whether sufficiently close operator algebras in a natural uniform sense must be small unitary perturbations of one another. For n≥3 and a free, ergodic, probability measure-preserving action of SL<sub>n</sub>(Z) on a standard nonatomic probability space (X,μ), write M=(L<sup>∞</sup>(X,μ)⋊SL<sub>n</sub>(Z))⊗¯¯¯R, where R is the hyperfinite II1-factor. We show that whenever M is represented as a von Neumann algebra on some Hilbert space H and N⊆B(H) is sufficiently close to M, then there is a unitary u on H close to the identity operator with uMu∗=N. This provides the first nonamenable class of von Neumann algebras satisfying Kadison and Kastler’s conjecture. We also obtain stability results for crossed products L<sup>∞</sup>(X,μ)⋊Γ whenever the comparison map from the bounded to usual group cohomology vanishes in degree 2 for the module L<sup>2</sup>(X,μ). In this case, any von Neumann algebra sufficiently close to such a crossed product is necessarily isomorphic to it. In particular, this result applies when Γ is a free group

A remark on the similarity and perturbation problems

In this note we show that Kadison's similarity problem for C*-algebras is equivalent to a problem in perturbation theory: must close C*-algebras have close commutants?Comment: 6 Pages, minor typos fixed. C. R. Acad. Sci. Canada, to appea

A characterization of operator algebras

AbstractAn operator algebra is a uniformly closed algebra of bounded operators on a Hilbert space. In this paper we give a characterization of unital operator algebras in terms of their matricial norm structure. More precisely if A is an L∞-matricially normed space and also an algebra with a completely contractive multiplication and an identity of norm 1, then there is a completely isometric isomorphism of A onto a unital operator algebra. Indeed the multiplication on A need not be assumed to be associative for this conclusion to follow. Examples are given to show that the condition on the identity is necessary. It follows from the above that the quotient of an operator algebra by a closed two-sided ideal (with the natural matricial structure) is again an operator algebra up to complete isometric isomorphism

C*-algebras nearly contained in type I algebras

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