On Vector-Valued Characters for Noncommutative Function Algebras

Abstract

Let A be a closed subalgebra of a C * -algebra, that is a closed algebra of Hilbert space operators. We generalize to such operator algebras A several key theorems and concepts from the theory of classical function algebras. In particular we consider several problems that arise when generalizing classical function algebra results involving characters ((contractive) homomorphisms into the scalars) on the algebra. For example, the Jensen inequality, the related Bishop-Ito-Schreiber theorem, and the theory of Gleason parts. We will usually replace characters (classical function algebra case) by D-characters, certain completely contractive homomorphisms Φ : A → D, where D is a C * -subalgebra of A. We also consider some D-valued variants of the classical Gleason-Whitney theorem.

Von Neumann algebra conditional expectations with applications to generalized representing measures for noncommutative function algebras

“…Then the normal extension ω to M = C * * is still a trace. As in the lines above Lemma 1.1 in [9], there is a central projection z ∈ M such that ω(z•) is a faithful normal tracial state on zM , and ω(zx) = ω(x) for all x ∈ M. So zM is a finite von Neumann algebra. Since the result which we are interested in is true for faithful normal tracial states, we have ω(|a| p ) = ω(z|a| p ) = ω(|za| p ) = ω(|a * z| p ), and similarly ω(|a * | p ) = ω(|a * z| p ).…”

“…We say that ω is an noncommutive Arens-Singer measure for Φ, if it satisfies this inequality for all invertible a ∈ A. These inequalities may be rewritten in terms of the ω-geometric mean ∆ ω (a) = exp(ω(log |a|)) (see the introduction to [9]). or example noncommutive Arens-Singer measures satisfy ∆ ω (Φ(a)) ≤ ∆ ω (a), a ∈ A −1 .…”

“…[15,33]). In [9], inspired by Arveson's seminal paper [1], we proposed a new generalization of characters applicable to operator algebras. By this last expression we mean a subalgebra A of a C * -algebra or von Neumann algebra M. In translating the classical theory of characters χ : A → C we may identify the range of the character with D = C1 A = C1 M , in which case the homomorphism becomes an idempotent map on A, and is a D-bimodule map.…”

“…Thus in the operator algebra case we suppose that we have a C * -(indeed in this paper usually von Neumann) subalgebra D of A and of M, and we define a D-character to be a unital contractive homomorphism Φ : A → D. We also assume that Φ is a D-bimodule map (or equivalently, is the identity map on D). The basic theory of D-characters is developed in [9]. One motivation for our study of such maps comes from [1], where Arveson gives many important examples of D-characters in his approach to noncommutative analyticity/generalized analytic functions/noncommutative Hardy spaces.…”

“…[10] for definitions-but this will not be important in the present paper. Indeed in the introduction to [9] we explain why D-characters are automatically completely contractive. By 4.2.9 in [10], which is an earlier result of the present authors, any unital completely contractive projection of a unital operator algebra onto a subalgebra D is a D-bimodule map.…”