Commutative C-algebras and Sequentially Normal Morphisms

We show that the image of a commutative monotone sequentially complete C∗-algebra, under a sequentially normal morphism, is again a monotone sequentially complete C-algebra, and also a monotone sequentially closed C∗-subalgebra. As a consequence, the image of an algebra of this type, under a sequentially normal representation in a separable Hilbert space, is strongly closed. In the case of a unital representation of C(X) in a separable Hilbert space, where X is a compact Hausdorff space, this implies that the von Neumann algebra generated by the image of C(X) is the image of the Baire functions on X under the extension of the representation to the bounded Borel functions. 1. Main result and application It is well-known that the image of a von Neumann algebra under a normal unital representation is again a von Neumann algebra [3, Theorem 2.5.3]. In this note, we prove a theorem in the same vein, but now in the category of C-algebras. The domain C-algebra is in our case a commutative algebra, possessing a sequential order completeness property to be defined below. The result is subsequently applied in the context of a unital separable representation of C(X), where X is a compact Hausdorff space. We start by recalling the relevant definitions. For the sake of clarity, let us mention explicitly that the C-algebras in this note are not necessarily unital; neither are (if applicable) the morphisms. Definition 1.1. Cf. [3, 3.9.2]. (1) A C-algebra A is monotone sequentially complete if every bounded increasing sequence of self-adjoint elements of A has a supremum in A. (2) A C-subalgebra A of a C-algebra B is a monotone sequentially closed C-subalgebra of B if supBn≥1 an ∈ A, whenever a1 ≤ a2 ≤ . . . is a bounded 2000 Mathematics Subject Classification. Primary 46L05; Secondary 46L10.