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# Journal of Operator Theory

Volume 59, Issue 1, Winter 2008  pp. 211-234.

Lipschitz and commutator estimates in symmetric operator spaces

Authors Denis Potapov (1) and Fyodor Sukochev (2)
Author institution: (1) School of Informatics and Engineering Faculty of Science and Engineering Flinders University of SA, Bedford Park, SA 5042 Australia
(2) School of Informatics and Engineering Faculty of Science and Engineering Flinders University of SA, Bedford Park, SA 5042 Australia

Summary:  This paper studies Lipschitz and commutator estimates in (non-commutative) symmetric operator spaces $\sE$ associated with a general semi-finite von Neumann algebra $\aM$ taken in its left regular representation. In particular, we show that if $f'$ is of bounded variation and $\sE$ is a reflexive (non-commutative) $L_p$-space on $\aM$, then the Lipschitz estimate $$\|f(a) - f(b)\|_\sE \leqslant c_f\, \|a - b\|_\sE,$$ holds for arbitrary self-adjoint operators $a$ and $b$ affiliated with $\aM$.

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