Operator algebra

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

Nonlocal (parafermion) currents in two-dimensional conformal quantum field theory and self-dual critical points in Z/sub N/-symmetric statistical systems

Abstract: A two-dimensional exactly solvable model of a conformal quantum field theory is developed which is self-dual and has Z/sub N/ symmetry. The operator algebra, the correlation functions, and the anomalous dimensions of all fields are calculated for this model, which describes self-dual critical points in Z/sub N/-symmetric statistical systems.

Regularity of rational vertex operator algebras

Abstract: A regular vertex operator algebra is a vertex operator algebra such that any weak module (without grading) is a direct sum of ordinary irreducible modules. In this paper we give several sufficient conditions under which a rational vertex operator algebra is regular. We prove that the moonshine module vertex operator algebra $V^{ atural},$ the vertex operator algebras $L(l,0)$ associated with the integrable representations of affine algebras of level $l,$ the vertex operator algebras $L(c_{p,q},0)$ associated with irreducible highest weight representations for the discrete series of the Virasoro algebra and the vertex operator algebras $V_L$ associated with positive definite even lattices $L$ are regular. Our result for $L(l,0)$ implies that any restricted integrable module of level $l$ for the corresponding affine Lie algebra is a direct sum of irreducible highest weight integrable modules. The space $V_L$ in general is a vertex algebra if $L$ is not positive definite. In this case we establish the complete reducibility of any weak module.

Notes on Compact Quantum Groups

Abstract: Compact quantum groups have been studied by several authors and from different points of view. The difference lies mainly in the choice of the axioms. In the end, the main results turn out to be the same. Nevertheless, the starting point has a strong influence on how the main results are obtained and on showing that certain examples satisfy these axioms. In these notes, we follow the approach of Woronowicz and we treat the compact quantum groups in the C ∗ -algebra framework. This is a natural choice when compact quantum groups are seen as a special case of locally compact quantum groups. A deep understanding of compact quantum groups in this setting is very important for understanding the problems that arise in developing a theory for locally compact quantum groups. We start with a discussion on locally compact quantum groups in general but mainly to motivate the choice of the axioms for the compact quantum groups. Then we develop the theory. We give the main examples and we show how they fit into this framework. The paper is an expository paper. It does not contain many new results although some of the proofs are certainly new and more elegant than the existing ones. Moreover, we have chosen to give a rather complete and self-contained treatment so that the paper can also serve as an introductory paper for non-specialists. Different aspects can be learned from these notes and a great deal of insight can be obtained. � Research Assistent of the National Fund for Scientific Research (Belgium)

Continuous Slice Functional Calculus in Quaternionic Hilbert Spaces

Abstract: The aim of this work is to define a continuous functional calculus in quaternionic Hilbert spaces, starting from basic issues regarding the notion of spherical spectrum of a normal operator. As properties of the spherical spectrum suggest, the class of continuous functions to consider in this setting is the one of slice quaternionic functions. Slice functions generalize the concept of slice regular function, which comprises power series with quaternionic coefficients on one side and that can be seen as an effective generalization to quaternions of holomorphic functions of one complex variable. The notion of slice function allows to introduce suitable classes of real, complex and quaternionic C*-algebras and to define, on each of these C*-algebras, a functional calculus for quaternionic normal operators. In particular, we establish several versions of the spectral map theorem. Some of the results are proved also for unbounded operators. However, the mentioned continuous functional calculi are defined only for bounded normal operators. Some comments on the physical significance of our work are included.