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.

Contents Full-Text PDF