Semi-discreteness and dilation theory for nest algebras

Abstract

In this paper it is shown that a contractive σ-weakly continuous Hilbert space representation of a nest algebra admits a σ-weakly continuous dilation to the containing algebra of all operators. This is accomplished by first establishing the complete contractivity of contractive representations through a semi-discreteness property for nest algebras relative to finite-dimensional nest algebras (Theorem 2.1). The semi-discreteness property is obtained by an examination of the order type, spectral type, and multiplicity of the nest, and by the construction of subalgebras that are completely isometric copies of finite-dimensional nest algebras, with good approximation properties. With complete contractivity at hand, the desired dilation follows from Arveson's dilation theorem and auxiliary arguments.