Journal of Operator Theory

Volume 54, Issue 1, Summer 2005 pp. 101-117.

$p$ -Summable commutators in dimension

$d$

Authors: William Arveson
Author institution: Department of Mathematics, University of California, Berkeley, CA 94720, USA

Summary: We show that many invariant subspaces

$M$ for

$d$ -shifts

$(S_1,\dots,S_d)$ of finite rank have the property that the orthogonal projection

$P_M$ onto

$M$ satisfies

$P_MS_k-S_kP_M\in\mathcal L^p,\quad 1\leqslant k\leqslant d$ for every

$p>2d$ ,

$\mathcal L^p$ denoting the Schatten-von Neumann class of all compact operators having

$p$ -summable singular value lists. In such cases, the

$d$ tuple of operators

$\overline T=(T_1,\dots,T_d)$ obtained by compressing

$(S_1,\dots,S_d)$ to

$M^\perp$ generates a

$*$ -algebra whose commutator ideal is contained in

$\mathcal L^p$ for every

$p>d$ . It follows that the

$C^*$ -algebra generated by

$\{T_1,\dots,T_d\}$ and the identity is commutative modulo compact operators, the Dirac operator associated with

$\overline T$ is Fredholm, and the index formula for the curvature invariant is stable under compact perturbations and homotopy for this restricted class of finite rank

$d$ -contractions. Though this class is limited, we conjecture that the same conclusions persist under much more general circumstances.

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