The Gelfand-Naimark theorem for C*-algebras over a ring of measurable functions

Abstract

One of the most important results in the theory of Banach algebras is the Gelfand–Naimark theorem. It describes a commutative C∗-algebra U over the field of complex numbers C as an algebra of continuous complex-valued functions defined on the set of pure states of U endowed with an ∗-weak topology. The development of the general theory of Banach–Kantorovich C∗-algebras over a ring of measurable functions naturally leads to the question about an analog of the Gelfand–Naimark theorem for such C∗-modules. The structural theory of C∗-modules originates from the work of I. Kaplansky [1], who used these objects in the algebraic approach to the theory of W ∗-algebras. Considering C∗-algebras, AW ∗-algebras, andW ∗-algebras as modules over their centers, one can describe some properties of the mentioned classes of ∗-algebras with the help of Boolean analysis methods (e.g., [2–5]). In particular, in [4] for C∗-algebras, representing modules over the Stone algebra, one obtains vector analogs of Gelfand–Mazur and Gleason–Zhelyazko–Kahan theorems. C∗-modules serve as useful examples of Banach–Kantorovich modules, whose theory is being actively developed now (e.g., [5, 6]). An efficient tool for the study of these Banach–Kantorovich modules, together with the Boolean-valued analysis, is the theory of continuous and measurable Banach bundles [6]. In particular, this enables one to represent a C∗-module over a ring of measurable functions as a measurable bundle of classical C∗-algebras [7]. This, in turn, allows one to establish properties ofC∗-modules, “gluing” the corresponding properties of C∗-algebras over the field C. In this paper we apply this approach, proving one version of the Gelfand– Naimark theorem for C∗-modules. We use the terminology and the notation of the theory of Banach– Kantorovich spaces [5] and the theory of measurable bundles stated in [6].