Operator algebra

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

The Classification Problem for Amenable C*-Algebras

Abstract: For thirty-five years, one of the most interesting and rewarding classes of operator algebras to study has been the approximately finite-dimensional C*-algebras of Glimm and Bratteli ([39], [7]).

On Schrödinger superalgebras

Abstract: Using the supersymplectic framework of Berezin, Kostant, and others, two types of supersymmetric extensions of the Schrodinger algebra (itself a conformal extension of the Galilei algebra) were constructed. An ‘I‐type’ extension exists in any space dimension, and for any pair of integers N+ and N−. It yields an N=N++N− superalgebra, which generalizes the N=1 supersymmetry Gauntlett et al. found for a free spin‐1/2 particle, as well as the N=2 supersymmetry of the fermionic oscillator found by Beckers et al. In two space dimensions, new, ‘exotic’ or ‘IJ‐type’ extensions arise for each pair of integers ν+ and ν−, yielding an N=2(ν++ν−) superalgebra of the type discovered recently by Leblanc et al. in nonrelativistic Chern–Simons theory. For the magnetic monopole the symmetry reduces to o(3)×osp(1/1), and for the magnetic vortex it reduces to o(2)×osp(1/2).

Renormalized Quantum Yang-Mills Fields in Curved Spacetime

Abstract: We present a proof that quantum Yang-Mills theory can be consistently defined as a renormalized, perturbative quantum field theory on an arbitrary globally hyperbolic curved, Lorentzian spacetime. To this end, we construct the non-commutative algebra of observables, in the sense of formal power series, as well as a space of corresponding quantum states. The algebra contains all gauge invariant, renormalized, interacting quantum field operators (polynomials in the field strength and its derivatives), and all their relations such as commutation relations or operator product expansion. It can be viewed as a deformation quantization of the Poisson algebra of classical Yang-Mills theory equipped with the Peierls bracket. The algebra is constructed as the cohomology of an auxiliary algebra describing a gauge fixed theory with ghosts and anti-fields. A key technical difficulty is to establish a suitable hierarchy of Ward identities at the renormalized level that ensure conservation of the interacting BRST-current, and that the interacting BRST-charge is nilpotent. The algebra of physical interacting field observables is obtained as the cohomology of this charge. As a consequence of our constructions, we can prove that the operator product expansion closes on the space of gauge invariant operators. Similarly, the renormalization group flow is proved not to leave the space of gauge invariant operators.

The Berezin transform and invariant differential operators

Abstract: The Berezin calculus is important to quantum mechanics (creation-annihilation operators) and operator theory (Toeplitz operators). We study the basic Berezin transform (linking the contravariant and covariant symbol) for all bounded symmetric domains, and express it in terms of invariant differential operators.

Differential Topology and Quantum Field Theory

Abstract: A Topological Preliminary. Elliptic Operators. Cohomology of Sheaves and Bundles. Index Theory for Elliptic Operators. Some Algebraic Geometry. Infinite Dimensional Groups. Morse Theory. Instantons and Monopoles. The Elliptic Geometry of Strings. Anomalies. Conformal Quantum Field Theories. Topological Quantum Field Theories. References.