From Semigroup Actions to the C∗-Algebra of Interaction Groups,

Abstract

author introduced an action θ of a sub-semigroup P of a group G and its corre-sponding action α of P, acting on the C∗-algebra C(X), consisting of all continuous functions, where X is a compact space. And the author found the corresponding unique interaction group. And the Toeplitz algebra and the C∗-algebra associated to this interaction group are defined. Even for the non-experts, this paper is easily-readable, since it is self-contained and well-written. In particular, the basic definitions and fundamental properties and the short history of interaction theory are well-introduced in Section 1 and 2. If one is interested in more about interaction theory, he or she can find the frontier works about interactions, and interaction groups, in [4], [5], and [6] (See Reference of this paper: Also see below). The one thing the reviewer wants to mention about is that: the author “just” introduced Toeplitz algebras and (crossed product) C∗-algebras associated to in-teraction groups in this paper. If the author provided more structure-theoretical properties or applications of them, the paper might be richer. Let A be a unital C∗-algebra and let ν, µ: A → A be maps. We say that the pair (ν, µ) of maps on A is an interaction, i