Schur multipliers of Cartan pairs

arXiv: Operator Algebras, 2020

In this paper we study the connection between Haagerup tensor product and compactness of Schur $A$-multiplier. In particular, we give a new characterization of elementary $C^{\ast}$-algebra in terms of completely compact Schur $A$-multiplier.

Proceedings of the Edinburgh Mathematical Society, 1994

For a large class of C'-algebras including all von Neumann algebras, the central Haagerup tensor product of the multiplier algebra with itself has an isometric representation as completely bounded operators.

MATHEMATICA SCANDINAVICA, 2003

Let $A\subset M$ be a MASA in a $\mathrm{II}_{1}$ factor $M$. We describe the von Neumann subalgebra of $M$ generated by $A$ and its normalizer $\mathcal N(A)$ as the set $N_q^w(A)$ consisting of those elements $m\in M$ for which the bimodule $\smash{\overline{AmA}}$ is discrete. We prove that two MASAs $A$ and $B$ are conjugate by a unitary $u\in N^{w}_{q}(A)$ iff $A$ is discrete over $B$ and $B$ is discrete over $A$ in the sense defined by Feldman and Moore [5]. As a consequence, we show that $A$ is a Cartan subalgebra of $M$ iff for any MASA $B\subset M$, $B=uAu^{*}$ for some $u\in M$ exactly when $A$ is discrete over $B$ and $B$ is discrete over $A$.

Russian Mathematical Surveys, 1971

The paper is concerned with von Neumann algebras with finite trace and their •-automorphisms, and with crossed products. A detailed investigation is made of the problem of constructing hyperfinite factors of type II t by means of crossed products. Some new results are obtained on subfactors of hyperfinite factors of type II ι and also some new information on the trajectory theory of measure-preserving transformations.

Eprint Arxiv Math 0609232, 2006

An (algebraic) extended bilinear Hilbert semispace H ∓ a is proposed as being the natural representation space for the algebras of von Neumann. This bilinear Hilbert semispace has a well defined structure given by the representation space Repsp(GL n (L v × L v )) of an (algebraic) complete bilinear semigroup GL n (L v × L v ) over the product of sets of completions characterized by increasing ranks.

K-Theory, 2000

To every von Neumann algebra one can associate a (multiplicative) determinant defined on the invertible elements of the algebra with range a subgroup of the abelian group of the invertible elements of the center of the von Neumann algebra. This determinant is a normalization of the usual determinant for finite von Neumann algebras of type I, for the type II 1-case it is the Fuglede-Kadison determinant, and for properly infinite von Neumann algebras the determinant is constant equal to 1. It is proved that every invertible element of determinant 1 is a product of a finite number of commutators. This extends a result of T. Fack and P. de la Harpe for II 1-factors. As a corollary it follows that the determinant induces an injection from the algebraic K 1-group of the von Neumann algebra into the abelian group of the invertible elements of the center. Its image is described. Another group, K w 1 (A), which is generated by elements in matrix algebras over A that induce injective right multiplication maps is also computed. We use the Fuglede-Kadison determinant to detect elements in the Whitehead group W h(G).

Mathematische Annalen, 2012

In this paper we present new structural information about the multiplier algebra M(A) of a σ-unital purely infinite simple C * -algebra A, by characterizing the positive elements A ∈ M(A) that are strict sums of projections belonging to A. If A ∈ A and A itself is not a projection, then the necessary and sufficient condition for A to be a strict sum of projections belonging to A is that A > 1 and that the essential norm A ess ≥ 1.

Bulletin of the American Mathematical Society

Editura Academiei, Bucure §ti, Romania, and Abacus Press, Turnbridge Wells, Kent, England, 1979, 478 pp.

Journal of Mathematical Physics, 2018

We consider the tensor product of the completely depolarising channel on d × d matrices with the map of Schur multiplication by a k × k correlation matrix and characterise, via matrix theory methods, when such a map is a mixed (random) unitary channel. When d = 1, this recovers a result of O'Meara and Pereira, and for larger d is equivalent to a result of Haagerup and Musat that was originally obtained via the theory of factorisation through von Neumann algebras. We obtain a bound on the distance between a given correlation matrix for which this tensor product is nearly mixed unitary and a correlation matrix for which such a map is exactly mixed unitary. This bound allows us to give an elementary proof of another result of Haagerup and Musat about the closure of such correlation matrices without appealing to the theory of von Neumann algebras.

Mathematische Annalen, 1997