A Geometric Characterization of Nonnegative Bands

http://dx.doi.org/10.4153/CMB-2004-025-0
Canad. Math. Bull. 47(2004), 257-263

  Published:2004-06-01
 Printed: Jun 2004

A band is a semigroup of idempotent operators. A nonnegative band $\cls$ in $\clb(\cll^2 (\clx))$ having at least one element of finite rank and with rank $(S) > 1 $ for all $S$ in $\cls$ is known to have a special kind of common invariant subspace which is termed a standard subspace (defined below). Such bands are called decomposable. Decomposability has helped to understand the structure of nonnegative bands with constant finite rank. In this paper, a geometric characterization of maximal, rank-one, indecomposable nonnegative bands is obtained which facilitates the understanding of their geometric structure.