Positive definite functions of Hopf $C\sp *$-algebras

Abstract

In this paper we study positive definite functions of Hopf C*- algebras. First of all, we introduce Fourier transformation on Hopf C* -algebras and use Fourier transform to characterize positive definite functions. Then we proceed to study smooth positive definite functions on Hopf C*-algebras. A complete description of smooth positive definite functions is obtained. Also, a Bochner type result for smooth positive definite functions is proved. 1. PRELIMINARIES In this section we first recall some of the terminology and results which we will need throughout this paper. Then we describe the contents of this paper. Let us start with the definitions of representations and the Peter-Weyl property for Hopf C*-algebras. Let (A, 0, k, e) be a Hopf C*-algebra with dense *-subalgebra -v. We say that it is involutive if k2 = I. For the definition and more information about Hopf C*-algebras, we refer to (WI) and (V). We start with the definition of comodules and representations of Hopf C*-algebras.