The Relative Nuclear Dimension of C*-Algebras, and the Nuclear Dimension of Generalised Toeplitz Algebras

Abstract

We consider the class of generalised Toeplitz algebras; those C*-algebras that can be expressed as an extension of C(X) by the compact operators K, for some compact metrizable space X. We show that one can generalise the result of Brake and Winter, that the nuclear dimension of the Toeplitz algebra is 1, to show that for any generalised Toeplitz algebra its nuclear dimension must be equal to dim_nuc C(X). This shows that Brake and Winter's dimension reduction phenomenon is applicable to a much wider class of algebras. We also introduce our definition for the relative nuclear dimension of a C*-algebra. This is a modification to the definition of nuclear dimension that requires us to factor through algebras of the form F \otimes B for F finite dimensional and B some fixed algebra we are working relative to. We explore various properties satisfied by the relative nuclear dimension with a particular eye to its being a modification of nuclear dimension.