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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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