Abelian strict approximation in $AW^*$-algebras and Weyl-von Neumann type theorems

Abstract:

In this paper, for a $C^*$-algebra $A$ with $M=M(A)$ an $AW^*$-algebra, or equivalently, for an essential, norm-closed, two-sided ideal $A$ of an $AW^*$-algebra $M$, we investigate the strict approximability of the elements of $M$ from commutative $C^*$-subalgebras of $A$. In the relevant case of the norm-closed linear span $A$ of all finite projections in a semi-finite $AW^*$-algebra $M$ we shall give a complete description of the strict closure in $M$ of any maximal abelian self-adjoint subalgebra (masa) of $A$. We shall see that the situation is completely different for discrete, respectively continuous, $M$:

In the discrete case, for any masa $C$ of $A$, the strict closure of $C$ is equal to the relative commutant $C’\cap M$, while in the continuous case, under certain conditions concerning the center valued quasitrace of the finite reduced algebras of $M$ (satisfied by all von Neumann algebras), $C$ is already strictly closed. Thus in the continuous case no elements of $M$ which are not already belonging to $A$ can be strictly approximated from commutative $C^*$-subalgebras of $A$.

In spite of this pathology of the strict topology in the case of the norm-closed linear span of all finite projections of a continuous semi-finite $AW^*$-algebra, we shall prove that in general situations also including this case, any normal $y\in M$ is equal modulo $A$ to some $x\in M$ which belongs to an order theoretical closure of an appropriate commutative $C^*$-subalgebra of $A$. In other words, if we replace the strict topology with order theoretical approximation, Weyl-von Neumann-Berg-Sikonia type theorems will hold in substantially greater generality.