Operator algebras and approximate diagonals

Diagonals in Tensor Products of Operator Algebras

In this paper we give a short, direct proof, using only properties of the Haagerup tensor product, that if an operator algebra A possesses a diagonal in the Haagerup tensor product of A with itself, then A must be isomorphic to a finite dimensional C∗-algebra. Consequently, for operator algebras, the first Hochschild cohomology group, H(A, X) = 0 for every bounded, Banach A-bimodule X , if and ...

One-sided Ideals and Approximate Identities in Operator Algebras

A left ideal of any C∗-algebra is an example of an operator algebra with a right contractive approximate identity (r.c.a.i.). Indeed, left ideals in C∗-algebras may be characterized as the class of such operator algebras, which happen also to be triple systems. Conversely, we show here and in a sequel to this paper, that operator algebras with r.c.a.i. should be studied in terms of a certain le...

Closed Projections and Approximate Identities for Operator Algebras

Let A be a (not necessarily selfadjoint) subalgebra of a unital C-algebra B which contains the unit of B. The right ideals of A with left contractive approximate identity are characterized as those subspaces of A supported by the orthogonal complement of a closed projection in B which also lies in A. Although this seems quite natural, the nonselfadjointness requires us to develop some interpola...

Operator Theory and Operator Algebras

Connes-amenability and normal, virtual diagonals for measure algebras

We prove that the measure algebra M(G) of a locally compact group G is Connesamenable if and only if G is amenable.