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' boolean AND 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 is an element of M is equal modulo A to some x. 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.
|Tipologia:||Articolo su rivista|
|Citazione:||D'Antoni, C., & Zsido, L. (2008). Abelian strict approximation in AW*-algebras and Weyl-von Neumann type theorems. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 360(9), 4705-4738.|
|Settore Scientifico Disciplinare:||Settore MAT/05 - Analisi Matematica|
|Revisione (peer review):||Sì, ma tipo non specificato|
|Digital Object Identifier (DOI):||10.1090/S0002-9947-08-04598-4|
|Stato di pubblicazione:||Pubblicato|
|Data di pubblicazione:||set-2008|
|Titolo:||Abelian strict approximation in AW*-algebras and Weyl-von Neumann type theorems|
|Autori interni:||ZSIDO, LASZLO |
|Autori:||D'Antoni, C; Zsido, L|
|Appare nelle tipologie:||01 - Articolo su rivista|
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