arXiv subjects
Publications and source records attributed to Roger Smith.
In this paper we introduce and study strongly singular maximal abelian self-adjoint subalgebras of type $II_1$ factors. We show that certain elements of free groups and of non-elementary hyperbolic groups generate such masas, and these also give new examples of masas for which Popa's invariant delta is 1. We also explore the connection between Popa's invariant and strong singularity.
The Laplacian (or radial) masa in a free group factor is generated by the sum of the generators and their inverses. We show that such a masa B is strongly singular and has Popa invariant delta(B) = 1. This is achieved by proving that the conditional expectation onto B is an asymptotic homomorphism. We also obtain similar results for the free product of discrete groups, each of which contains an element of infinite order.
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, $H^1(A,X) = 0$ for every bounded, Banach A-bimodule X, if and only if A is isomorphic to a finite dimensional $C^*$-algebra.