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# Journal of Operator Theory

Volume 45, Issue 2, Spring 2001  pp. 303-334.

Differential Schatten $*$-algebras. Approximation property and approximate identities

Authors Edward Kissin (1), and Victor S. Shulman (2)
Author institution: (1) School of Mathematical Sciences, University of North London, Holloway, London N7 8DB, G.B.
(2) Department of Mathematics, Vologda State Technical Uni versity, Vologda, Russia

Summary:  For symmetric operators $S$, we consider differential Schatten algebras $C_{S }^{p,q}$ of compact operators $A$ from $C^{p}$ with $SA-AS$ belonging to $C^{q}$. Thes e algebras are analogues of the Sobolev $W_{p,q}^{1}$ spaces. We study their approximation pr operty: whether every operator is approximated by finite rank operators, and the existence of approximate identities. For non-selfadjoint $S$, we show that $C _{S}^{p,q}$ have no bounded approximate identities and the product of any two operators is approximated by finite rank operators. For selfadjoint $S$, $C_{S}^{p,q}$ have approximate identities consisting of finite rank operators and hence, have the approximation property. These identities are bounded only if $p = \infty$. The existence of a bounded identity for $C_{S}^{\infty,1}$ is equivalent to $1$- semidiagonality of $S$.

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