OFF-DIAGONAL CORNERS OF SUBALGEBRAS OF L(Cn)

Abstract

Let n ∈ N, and consider Cn equipped with the standard inner product. Let A ⊆ L(Cn) be a unital algebra and P ∈ L(Cn) be an orthogonal projection. The space L := P ⊥A|ran P is said to be an off-diagonal corner of A, and L is said to be essential if ∩{ker L : L ∈ L} = {0} and ∩{ker L ∗ : L ∈ L} = {0}, where L ∗ denotes the adjoint of L. Our goal in this paper is to determine effective upper bounds on dim A in terms of dim L, where L is an essential off-diagonal corner of A. A detailed structure analysis of A based upon the dimension of L, while seemingly elusive in general, is nevertheless provided in the cases where dim L ∈ {1, 2}.