Operator algebra

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

Operator space theory: a natural framework for bell inequalities.

Abstract: In this letter we show that the field of Operator Space Theory provides a general and powerful mathematical framework for arbitrary Bell inequalities, in particular regarding the scaling of their violation within quantum mechanics. We illustrate the power of this connection by showing that bipartite quantum states with local Hilbert space dimension n can violate a Bell inequality by a factor of order $\frac{\sqrt{n}}{\log^2n}$ when observables with n possible outcomes are used. Applications to resistance to noise, Hilbert space dimension estimates and communication complexity are given.

Lifting differential operators from orbit spaces

Abstract: Let X be an affine complex algebraic variety, and let T>(X) denote the (non-commutative) algebra of algebraic differential operators on X. Then T>(X) has a filtration {^(X)} by order of differentiation, and the associated graded grT>(X) is commutative. Now assume that X is smooth and a Gvariety, where G is a reductive complex algebraic group. Let TTX : X —> X//G be the quotient morphism. Then we have a natural map (Ti-x)* : (^(X)) —> ^(X/fG). We find conditions under which (Tvx)* is surjective for all n, in which case grD(X//G) is finitely generated. We conjecture that the latter is always true. We also consider generalizations to algebras of differential operators on sections of Gvector bundles.

Algebras of modal operators and partial correctness

Abstract: Modal Kleene algebras are Kleene algebras enriched by forward and backward box and diamond operators. We formalise the symmetries of these operators as Galois connections, complementarities and dualities. We study their properties in the associated operator algebras and show that the axioms of relation algebra are theorems at the operator level. Modal Kleene algebras provide a unifying semantics for various program calculi and enhance efficient cross-theory reasoning in this class, often in a very concise pointfree style. This claim is supported by novel algebraic soundness and completeness proofs for Hoare logic and by connecting this formalism with an algebraic decision procedure.

A non-commutative framework for topological insulators

Abstract: We study topological insulators, regarded as physical systems giving rise to topological invariants determined by symmetries both linear and anti-linear. Our perspective is that of non-commutative index theory of operator algebras. In particular, we formulate the index problems using Kasparov theory, both complex and real. We show that the periodic table of topological insulators and superconductors can be realized as a real or complex index pairing of a Kasparov module capturing internal symmetries of the Hamiltonian with a spectral triple encoding the geometry of the sample’s (possibly non-commutative) Brillouin zone.

The basis of nonlocal curvature invariants in quantum gravity theory. Third order.

Abstract: A complete basis of nonlocal invariants in quantum gravity theory is built to third order in space–time curvature and matter‐field strengths. The nonlocal identities are obtained which reduce this basis for manifolds with dimensionality 2ω<6. The present results are used in heat‐kernel theory, theory of gauge fields and serve as a basis for the model‐independent approach to quantum gravity and, in particular, for the study of nonlocal vacuum effects in the gravitational collapse problem.