C-ASTERISK-CONVEXITY AND MATRIC RANGES

C*-convex sets in matrix algebras are convex sets of matrices in which matrix-valued convex coefficients are admitted along with the usual scalar-valued convex coefficients. A Caratheodory-type theorem is developed for C*-convex hulls of compact sets of matrices, and applications of this theorem are given to the theory of matricial ranges. If T is an element in a unital C*-algebra A, then for every n is-an-element-of N, the n x n matricial range W(n)(T) of T is a compact C*-convex set of n x n matrices. The basic relation W1(T) = conv sigma(T) is well known to hold if T exhibits the normal-like quality of having the spectral radius of beta-T + mu-1 coincide with the norm parallel-to beta-T + mu-1 parallel-to for every pair of complex numbers beta and mu. An extension of this relation to the matrix spaces is given by Theorem 2.6: W(n)(T) is the C*-convex hull of the n x n matricial spectrum sigma(n)(T) of T if, for every B, M is-an-element-of M(n), the norm of T x B + 1 x M in A x M(n) is the maximum value in {parallel-to LAMBDA x B + 1 x M parallel-to : LAMBDA is-an-element-of sigma(n)(T)}. The spatial matricial range of a Hilbert space operator is the analogue of the classical numerical range, although it can fail to be convex if n > 1. It is shown in sectional sign 3 that if T has a normal dilation N with sigma(N) subset-of sigma(T), then the closure of the spatial matricial range of T is convex if and only if it is C*-convex.