Journal of Operator Theory

Volume 55, Issue 2, Spring 2006 pp. 225-238.

Sectional curvature and commutation of pairs of selfadjoint operators

Authors: E. Andruchow (1) and L. Recht (2)
Author institution: (1) Instituto de Ciencias, Universidad Nacional de General Sarmiento, Los Polvorines, 1613, Argentina
(2) Departamento de Matemáticas Puras y Aplicadas, Universidad Simón Bolívar, Caracas, 1080A, Venezuela

Summary: The space $\mathcal{G}^+$ of postive invertible operators of a $C^*$-algebra $\mathcal{A}$, with the appropriate Finsler metric, behaves like a (non positively curved) symmetric space. Among the characteristic properties of such spaces, one has that two selfadjoint elements $x,y\in\mathcal{A}$ $($regarded as tangent vectors at $a\in \mathcal{G}^+$$)$ verify that \begin{equation*} \|x-y\|_a\leqslant d(\mathrm{exp}_a(x),\mathrm{exp}_a(y)). \end{equation*} In this paper we investigate the ocurrence of the equality \begin{equation*} \|x-y\|_a=d(\mathrm{exp}_a(x),\mathrm{exp}_a(y)). \end{equation*} If $\mathcal{A}$ has a trace, and the trace is used to measure tangent vectors then, as in the finite dimensional classical setting, this equality is equivalent to the fact that $x$ and $y$ commute. In arbitrary $C^*$-algebras, when the usual $C^*$-norm is used, the equality is equivalent to a weaker condition. We introduce in $\mathcal{G}^+$ an analogous of the sectional curvature for pairs of selfadjoint operators, and study the vanishing of this invariant.

Keywords: Positive operator, selfadjoint operator, sectional curvature.

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