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A Geometric Characterization of Nonnegative Bands

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.