The properties *-regularity and uniqueness of 𝐶*-norm in a general *-algebra

Abstract

Abstract. In this paper two properties of a "-algebra A are considered which are concerned with the relationship between the algebra and its C*-enveloping algebra. These properties are that A have a unique C*-norm, and that A be '-regular. Both of these concepts are closely involved with the representation theory of the algebra.

Polynomial growth of discrete quantum groups, topological dimension of the dual and *-regularity of the Fourier algebra

“…It is known that A has a unique C * -norm if and only if I ∩ A = 0 for every nonzero ideal I as above and this implies that * -regular Banach algebras have a unique C *norm. Furthermore, A is * -regular if and only if all quotients A/A ∩ I have a unique C * -norm [5].…”

“…If A is such an algebra, an equivalent statement of * -regularity is that every closed ideal of C * (A), has dense intersection with A. As an important general consequence of this property we have that A has a unique C * -norm [4,5,8]. This in particular applies to A(G), as opposed to the canonical dense Hopf algebra which has many C * -norms already for G = T. The property of * -regularity has been first shown for the pair of the group algebra L 1 (Γ) of a locally compact group with Haar measure of polynomial growth and its C * -envelope C * (Γ) [7].…”

“…As a matter of fact, it is not as easy to find examples among amenable discrete groups. Indeed, on the one hand the relationship between * -regularity and C * -uniqueness has been studied at an in-depth level in [5]. On the other, it is not known whether every such group is automatically C * -unique, see [20] for partial positive results.…”