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Ranks of Algebras of Continuous C*-Algebra Valued Functions

Published online by Cambridge University Press:  20 November 2018

Masaru Nagisa ,
Hiroyuki Osaka  and
N. Christopher Phillips
Masaru Nagisa
Affiliation:
Department of Mathematics and Informatics, Chiba University, Yayoi-cho 1-33, Inage-ku, Chiba City 263-8522, Japan
Hiroyuki Osaka
Affiliation:
Department of Mathematical Sciences, Ritsumeikan University, Kusatsu, Shiga, 525-8577, Japan
N. Christopher Phillips
Affiliation:
Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222, USA
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Abstract

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We prove a number of results about the stable and particularly the real ranks of tensor products of ${{C}^{*}}$ -algebras under the assumption that one of the factors is commutative. In particular, we prove the following:

  1. (1) If $X$ is any locally compact $\sigma $-compact Hausdorff space and $A$ is any ${{C}^{*}}$-algebra, then $\text{RR(}{{C}_{0}}\text{(}X\text{)}\otimes A\text{)}\le \text{dim(}X\text{)+RR(}A\text{)}$ .

  2. (2) If $X$ is any locally compact Hausdorff space and $A$ is any purely infinite simple ${{C}^{*}}$ -algebra, then $\text{RR(}{{C}_{0}}\text{(}X\text{)}\otimes A\text{)}\le 1$ .

  3. (3) $\text{RR(}C([0,\,1]\,)\otimes \,A)\,\ge \,1$ for any nonzero ${{C}^{*}}$-algebra $A$, and $\text{sr(}C({{[0,\,1]}^{2}})\,\otimes \,A\text{)}\,\ge \,2$ for any unital ${{C}^{*}}$-algebra $A$.

  4. (4) If $A$ is a unital ${{C}^{*}}$-algebra such that $\text{RR(}A\text{)}\,\text{=}\,\text{0,}\,\text{s}r\text{(}A\text{)}\,\text{=}\,\text{1}$, and ${{K}_{1}}(A)=0$ , then $\text{sr(}C([0,\,1])\,\otimes \,A\text{)}\,\text{=}\,1$.

  5. (5) There is a simple separable unital nuclear ${{C}^{*}}$-algebra $A$ such that $\text{RR(}A\text{)}\,\text{=}\,\text{1}$ and $\text{sr(}C([0,\,1])\,\otimes \,A\text{)}\,\text{=}\,1$.

Keywords

46L0546L5246L8019A1319B10
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 53 , Issue 5 , 01 October 2001 , pp. 979 - 1030
Copyright
Copyright © Canadian Mathematical Society 2001

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