Operator algebra

About: Operator algebra is a research topic. Over the lifetime, 5783 publications have been published within this topic receiving 165303 citations.

Uniform algebra isomorphisms and peripheral multiplicativity

Abstract: Let φ: A→ B be a surjective operator between two uniform alge-(1) =1 we Show that if φ satisfies the peripheral multiplicativity bras with φ(1) = 1. We show that if if satisfies the peripheral multiplicativity condition σ π (φ(f)φ(g)) = σ π (fg) for all f,g ∈ A, where σ π (f) is the peripheral spectrum of f, then φis an isometric algebra isomorphism from A onto B. One of the consequences of this result is that any surjective, unital, and multiplicative operator that preserves the peripheral ranges of algebra elements is an isometric algebra isomorphism. We describe also the structure of general, not necessarily unital, surjective and peripherally multiplicative operators between uniform algebras.

Reduced Operator Algebras of Trace-Preserving Quantum Automorphism Groups

Abstract: Let B be a finite dimensional C ∗ -algebra equipped with its canonical trace induced by the regular representation of B on it- self. In this paper, we study various properties of the trace-preserving quantum automorphism group G of B. We prove that the discrete dual quantum group b G has the property of rapid decay, the reduced von Neumann algebra L ∞ (G) has the Haagerup property and is solid, and that L ∞ (G) is (in most cases) a prime type II1-factor. As applica- tions of these and other results, we deduce the metric approximation property, exactness, simplicity and uniqueness of trace for the reduced C ∗ -algebra Cr(G), and the existence of a multiplier-bounded approx- imate identity for the convolution algebra L 1 (G).

Quantum energy inequalities and local covariance II: Categorical formulation

Abstract: We formulate quantum energy inequalities (QEIs) in the framework of locally covariant quantum field theory developed by Brunetti, Fredenhagen and Verch, which is based on notions taken from category theory. This leads to a new viewpoint on the QEIs, and also to the identification of a new structural property of locally covariant quantum field theory, which we call local physical equivalence. Covariant formulations of the numerical range and spectrum of locally covariant fields are given and investigated, and a new algebra of fields is identified, in which fields are treated independently of their realisation on particular spacetimes and manifestly covariant versions of the functional calculus may be formulated.