Pseudo-B-Fredholm operators, poles of the resolvent and mean convergence in the calkin algebra

We define here a pseudo B-Fredholm operator as an operator such that 0 is isolated in its essential spectrum, then we prove that an operator T is pseudo-B-Fredholm if and only if T = R + F where R is a Riesz operator and F is a B-Fredholm operator such that the commutator [R,F] is compact. Moreover, we prove that 0 is a pole of the resolvent of an operator T in the Calkin algebra if and only if T = K + F, where K is a power compact operator and F is a B-Fredholm operator, such that the commutator [K,F] is compact. As an application, we characterize the mean convergence in the Calkin algebra.

SAW*-ALGEBRAS ARE ESSENTIALLY NON-FACTORIZABLE

In this paper, we solve a question of Simon Wassermann, whether the Calkin algebra can be written as a C*-tensor product of two infinite dimensional C*-algebras. More generally, we show that there is no surjective *-homomorphism from a SAW*-algebra onto C*-tensor product of two infinite dimensional C*-algebras.