Ranks of algebras of continuous C*-algebra valued functions (Q2753432)

In the paper under review the real rank of tensor products of \(C^\ast\)-algebras is analyzed provided that one of the factors is abelian. Let \(RR(A)\) and \(SR(A)\) denote the real rank and stable rank of a \(C^\ast\)-algebra \(A\), respectively. Many interesting inequalities for ranks of tensor products are established. Among others, the following results are obtained: NEWLINE\[NEWLINE RR(C_0(X)\otimes A)\leq\dim(X)+RR(A),NEWLINE\]NEWLINE whenever \(X\) is a locally compact \(\sigma\)-compact Hausdorff space and \(A\) is any \(C^\ast\)-algebra; NEWLINE\[NEWLINERR(C_0(X)\otimes A)\leq 1,NEWLINE\]NEWLINE whenever \(X\) is a locally compact Hausdorff space and \(A\) is any purely infinite simple \(C^\ast\)-algebra; NEWLINE\[NEWLINERR(C[0,1]\otimes A)\geq 1, NEWLINE\]NEWLINE whenever \(A\) is nonzero, and NEWLINE\[NEWLINE SR(C[0,1]^2\otimes A)\geq 2,NEWLINE\]NEWLINE whenever \(A\) is a unital \(C^\ast\)-algebra.