Exponential Rank and Exponential Length of Operators on Hilbert C * - Modules

Abstract

In this article we consider unitary operators on the Hilbert C*-module HA associated with a large class of C*-algebras. Let L(X) denote the algebra consisting of all bounded linear operators on a separable Hilbert space 7 and let K denote the algebra of all compact operators on R. Halmos and Kakutani proved that every unitary operator on R is a product of four symmetries (cf. [Ha], 143; [HaKa]). For the case when X is a compact Hausdorff space, Ringrose [R] recently considered the C*-algebra C(X, L(Q)), consisting of all norm-continuous maps from X to L(X), and proved that every unitary element of C(X, L(X)) can be written as a product of, at most, three exponentials; in other words, a norm-continuous map u(.) from X to the unitary group of L(X) forms a product of three norm-continuous maps written as exp(ihi) exp(ih2) exp(ih3), where hl, h2 and h3 are maps from X to the space of self-adjoint operators in L(X). Phillips and Ringrose [PhR] conjectured that every unitary of C(X, L(X)) can even be approximated in the norm by products of two exponentials. Using techniques in the theory of C*-algebras, but by a different approach, we generalized Ringrose's result to Hilbert C*-modules associated with all a-unital (in particular, all unital) C*-algebras (cf. [Z8]). We also proved in [Z8] similar results for the C*-algebra consisting of norm-continuous maps from X to a purely infinite simple C*-algebra. The purpose of this article is to settle the Phillips-Ringrose conjecture in the affirmative and to prove a generalized version of the Halmos-Kakutani theorem in [HaKa]. We consider the group of unitary operators on Hilbert C*-modules associated with the following two specific classes, which together comprise the majority of interesting C*-algebras: