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Goldie dimension for C*-algebras

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  11 January 2024

Mohammad Rouzbehani Open the ORCID record for Mohammad Rouzbehani [Opens in a new window] ,
Massoud Amini Open the ORCID record for Massoud Amini [Opens in a new window]  and
Mohammad B. Asadi
Mohammad Rouzbehani
Affiliation:
School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran (rouzbehani.m.math@gmail.com, mb.asadi@ut.ac.ir)
Massoud Amini
Affiliation:
Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran 14115-134, Iran (mamini@modares.ac.ir) School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran 19395-5746, Iran (mamini@ipm.ir)
Mohammad B. Asadi
Affiliation:
School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran (rouzbehani.m.math@gmail.com, mb.asadi@ut.ac.ir)
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Abstract

In this article, we introduce and study the notion of Goldie dimension for C*-algebras. We prove that a C*-algebra A has Goldie dimension n if and only if the dimension of the center of its local multiplier algebra is n. In this case, A has finite-dimensional center and its primitive spectrum is extremally disconnected. If moreover, A is extending, we show that it decomposes into a direct sum of n prime C*-algebras. In particular, every stably finite, exact C*-algebra with Goldie dimension, that has the projection property and a strictly full element, admits a full projection and a non-zero densely defined lower semi-continuous trace. Finally we show that certain C*-algebras with Goldie dimension (not necessarily simple, separable or nuclear) are classifiable by the Elliott invariant.

Keywords

C*-algebralocal multiplier algebragraph C*-algebraArtiniancompact idealpurely infiniteElliott invariant

MSC classification

Primary: 46L05: General theory of $C^*$-algebras 46L80: $K$-theory and operator algebras (including cyclic theory)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 67 , Issue 1 , February 2024 , pp. 200 - 223
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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