John Lindsay Orr

ON THE CLOSURE OF TRIANGULAR ALGEBRAS

algebras of operators on Hilbert space introduced by Kadison and Singer in [8]. If sA is a maximal abelian self-adjoint algebra (m.a.s.a.) in B(NJ) then an algebra of operators, 9T, is said to be triangular with diagonal, S, if S-T n o* = s4. This definition transfers naturally to a subalgebra, T, of a von Neumann algebra, M, for which T n T* is a m.a.s.a. in M. Recently, Muhly, Saito and Solel [9] have made an extensive study of a-weakly closed triangular subalgebras of von Neumann algebras when the diagonal m.a.s.a. is a Cartan subalgebra of M. In particular, they have obtained a structure theorem for cr-weakly closed maximal triangular algebras which generalizes the most regular behaviour of triangular algebras in B(NW). However, the most general results on triangular algebras (by which we mean results for which as little as possible is assumed about the algebra beyond its triangularity) have proved elusive. One natural question of this type, which was raised by Erdos in [5], is; Question 1. Is the norm closure of a triangular algebra necessarily triangular? It follows from a straightforward Zorns Lemma argument that every triangular algebra in B(C) (resp. a von Neumann algebra, M) is contained in a maximal triangular algebra. Thus, Question 1 is equivalent to,

EPIMORPHISMS OF NEST ALGEBRAS

We study epimorphisms of one nest algebra onto another. Such maps are always continuous. Most are obtained from a compression to an interval followed by a similarity. The others factor through an ultraproduct of the discrete intervals of finite rank atoms. When the image nest has no infinite part, these maps are limits of compression maps along an ultrafilter.

An estimate on the norm of the product of infinite block operator matrices

Abstract A lower bound is found for the operator norm of the product of two infinite block operator matrices on Hilbert space which depends only on the norms of submatrices of these operators. The methodology of the paper is to study sequence-valued functions on certain families of finite sets (the barriers of Nash-Williams) and show that the convex halls of the ranges of any pair of such functions must contain points which are close. This is achieved by a study in infinite Ramsey theory.

A NOTE ON QUASICENTRAL APPROXIMATE UNITS IN B(H)

If a Hubert Space, H , is infinite dimensional, B{H) has no count- able quasicentral approximate unit for the ideal of finite rank operators.