Abstract view
Ranks of Algebras of Continuous $C^*$Algebra Valued Functions


Published:20011001
Printed: Oct 2001
Masaru Nagisa
Hiroyuki Osaka
N. Christopher Phillips
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Abstract
We prove a number of results about the stable and particularly the
real ranks of tensor products of \ca s under the assumption that one
of the factors is commutative. In particular, we prove the following:
{\raggedright
\begin{enumerate}[(5)]
\item[(1)] If $X$ is any locally compact $\sm$compact Hausdorff space
and $A$ is any \ca, then\break
$\RR \bigl( C_0 (X) \otimes A \bigr) \leq
\dim (X) + \RR(A)$.
\item[(2)] If $X$ is any locally compact Hausdorff space and $A$ is
any \pisca, then $\RR \bigl( C_0 (X) \otimes A \bigr) \leq 1$.
\item[(3)] $\RR \bigl( C ([0,1]) \otimes A \bigr) \geq 1$ for any
nonzero \ca\ $A$, and $\sr \bigl( C ([0,1]^2) \otimes A \bigr) \geq 2$
for any unital \ca\ $A$.
\item[(4)] If $A$ is a unital \ca\ such that $\RR(A) = 0$, $\sr (A) =
1$, and $K_1 (A) = 0$, then\break
$\sr \bigl( C ([0,1]) \otimes A \bigr) = 1$.
\item[(5)] There is a simple separable unital nuclear \ca\ $A$ such
that $\RR(A) = 1$ and\break
$\sr \bigl( C ([0,1]) \otimes A \bigr) = 1$.
\end{enumerate}}
MSC Classifications: 
46L05, 46L52, 46L80, 19A13, 19B10 show english descriptions
General theory of $C^*$algebras Noncommutative function spaces $K$theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22] Stability for projective modules [See also 13C10] Stable range conditions
46L05  General theory of $C^*$algebras 46L52  Noncommutative function spaces 46L80  $K$theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22] 19A13  Stability for projective modules [See also 13C10] 19B10  Stable range conditions
