# Journal of Operator Theory

Volume 47, Issue 1, Winter 2002 pp. 117-130.

Real rank and exponential length of tensor products with $\Cal O_\infty$**Authors**: N. Christopher Phillips

**Author institution:**Department of Mathematics, University of Oregon, Eugene, OR 97403--1222, USA

**Summary:**Let $D$ be any \ca. We prove that $\Cal O_\infty \otimes D$ has real rank at most $1$, exponential length at most $2 \pi$, exponential rank at most $2 + \varepsilon$, and $C^*$ projective length at most $\pi$. The algebra $\Cal O_\infty$ can be replaced with any separable nuclear purely infinite simple \ca.

**Keywords:**Real rank, exponential rank, exponential length, projective length, in finite $C^*$-algebras

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