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


Authors: Claudio D'Antoni and László Zsidó
Journal: Trans. Amer. Math. Soc. 360 (2008), 4705-4738
MSC (2000): Primary 46L05; Secondary 46L10
DOI: https://doi.org/10.1090/S0002-9947-08-04598-4
Published electronically: April 7, 2008
MathSciNet review: 2403702
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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.


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Additional Information

Claudio D'Antoni
Affiliation: Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy
Email: dantoni@axp.mat.uniroma2.it

László Zsidó
Affiliation: Dipartimento di Matematica, Università di Roma “Tor Vergata” Via della Ricerca Scientifica, 00133 Roma, Italy
Email: zsido@axp.mat.uniroma2.it

DOI: https://doi.org/10.1090/S0002-9947-08-04598-4
Received by editor(s): June 19, 2006
Published electronically: April 7, 2008
Additional Notes: This work was supported by the MIUR, INDAM and EU
Dedicated: Dedicated to Professor E. Effros on his $ 70^{th}$ birthday
Article copyright: © Copyright 2008 American Mathematical Society

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