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Journal of Operator Theory

Volume 71, Issue 1, Winter 2014  pp. 175-197.

Asymptotic commutativity

Authors: Mihai Sabac
Author institution:Department of Mathematics, University of Bucharest, Str. Academiei 14, RO-70109 Bucharest, Romania

Summary:  The standard commutants in a noncommutative algebra are derived from commutativity which in terms of Lie algebras means $(\mathrm{ad}T)(S)=0$. Some `weaker commutativities'' given by vanishing (asymptotic vanishing) properties of the powers of $\mathrm{ad}T$, for instance $(\mathrm{ad}T)^{n}(S)=0$ or\break $\lim\limits_{n\rightarrow\infty}\|(\mathrm{ad}T)^{n}(S)\|^{1/n}$ $=0$ when $T$ and $S$ are bounded linear operators on some complex Banach space, describe in a similar way different type of `weaker commutants''. This paper studies these `weaker commutants'' and their corresponding compositions, in particular `weaker bicommutants", in connection with J.~von Neumann's classical bicommutant theorem.

Keywords:  bounded operator, commutant, selfadjoint operator algebra.

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