Journal of Operator Theory

Volume 48, Issue 3, Supplementary 2002 pp. 633-643.

Locally minimal projections in $B(H)$

Authors: Charles A. Akemann (1), and Joel Anderson (2)
Author institution: (1) Department of Mathematics, University of California, Santa Barbara, CA 93106, USA
(2) Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA

Summary: Given an $n$-tuple $\{a_1, \ldots, a_n\}$ of self-adjoint operators on an infinite dimensional Hilbert space $H$, we say that a projection $p$ in $B(H)$ is {\it locally minimal} for $\{a_1, \ldots, a_n\}$ if each $pa_jp$, for $j = 1, \ldots, n$, is a scalar multiple of $p$. In Theorem 1.8 we show that for any such $\{a_1, \ldots, a_n\}$ and any positive integer $k$ there exists a projection $p$ of rank $k$ that is locally minimal for $\{a_1, \ldots, a_n\}$. If we further assume that $\{a_1,\ldots,a_n,1\}$ is a linearly independent set in the Calkin algebra, then in Theorem 2.10 we prove that $p$ can be chosen of infinite rank.

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