Operator algebra

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

Discrete product systems of Hilbert bimodules

Abstract: A Hilbert bimodule is a right Hilbert module X over a C*-algebra A together with a left action of A as adjointable operators on X. We consider families X = {X s : s E P} of Hilbert bimodules, indexed by a semigroup P, which are endowed with a multiplication which implements isomorphisms X s ⊗ A X t → X st ; such a family is a called a product system. We define a generalized Cuntz-Pimsner algebra O x , and we show that every twisted crossed product of A by P can be realized as O x for a suitable product system X. Assuming P is quasi-lattice ordered in the sense of Nica, we analyze a certain Toeplitz extension T cov (X) of O X by embedding it in a crossed product B P A T,X P which has been twisted by X; our main Theorem is a characterization of the faithful representations of B P X T,X P.

On the multiplication of free N-tuples of noncommutative random variables

Abstract: Let a1, , a n , b 1 , , b n be random variables in a noncommutative probability space, such that a1, , a n is free from b1, , b n . We show how the joint distribution of the n-tuple (a1b1, , a n b n ) can be described in terms of the joint distributions of (a1, , a n )a nd ( b 1 , , b n ), by using the combinatorics of the n-dimensional R-transform. We point out a few applications that can be easily derived from our result, concerning the left-and-right translation with a semicircular element (see Sections 1.6-1.10) and the compression with a projection (see Sections 1.11-1.14) of an n-tuple of noncommutative random variables. A different approach to two of these applications is presented by Dan Voiculescu in an Appendix to the paper. Introduction. The theory of free random variables was developed in a se- quence of papers of D. Voiculescu (see (17), or the recent survey in (16)), as an instrument for approaching free products of operator algebras. Its particular aspect addressed in the present paper is the one concerning the addition and multiplica- tion of free random variables; as shown by Voiculescu in (13), (14), a powerful method in the study of these operations is the use of transforms that convert them (respectively) into addition and multiplication of complex analytic func- tions, or, in an algebraic framework, of formal power series in an indeterminate z. The precise definitions of these transforms (called R-transform for the addition problem and S-transform for the multiplication problem) will be reviewed in the Sections 1.2, 1.3 below. In the present paper we are pursuing a combination of two ideas that have appeared recently in the study of the R- and S-transforms. The first idea is that the connection between the R- and the S-transform is closer than one might suspect at first glance. A way of making this precise was pointed out in our paper (6), in the form of the equation S( )= ( R ( )) for a distribution with nonvan- ishing mean, and where is a combinatorial object with a precise significance ("the Fourier transform for multiplicative functions on noncrossing partitions"). A byproduct of our result in (6) is the remark that the multiplication of free

Model theory of operator algebras II: model theory

Abstract: We introduce a version of logic for metric structures suitable for applications to C*-algebras and tracial von Neumann algebras. We also prove a purely model-theoretic result to the effect that the theory of a separable metric structure is stable if and only if all of its ultrapowers associated with nonprincipal ultrafilters on ℕ are isomorphic even when the Continuum Hypothesis fails.

Vertex operator algebras, Higgs branches, and modular differential equations

Abstract: Every four-dimensional $$ \mathcal{N}=2 $$ superconformal field theory comes equipped with an intricate algebraic invariant, the associated vertex operator algebra. The relationships between this invariant and more conventional protected quantities in the same theories have yet to be completely understood. In this work, we aim to characterize the connection between the Higgs branch of the moduli space of vacua (as an algebraic geometric entity) and the associated vertex operator algebra. Ultimately our proposal is simple, but its correctness requires the existence of a number of nontrivial null vectors in the vacuum Verma module of the vertex operator algebra. Of particular interest is one such null vector whose presence suggests that the Schur index of any $$ \mathcal{N}=2 $$ SCFT should obey a finite order modular differential equation. By way of the “high temperature” limit of the superconformal index, this allows the Weyl anomaly coefficient a to be reinterpreted in terms of the representation theory of the associated vertex operator algebra. We illustrate these ideas in a number of examples including a series of rank-one theories associated with the “Deligne-Cvitanovic exceptional series” of simple Lie algebras, several families of Argyres-Douglas theories, an assortment of class $$ \mathcal{S} $$ theories, and $$ \mathcal{N}=2 $$ super Yang-Mills with $$ \mathfrak{s}\mathfrak{u}(n) $$ gauge group for small-to-moderate values of n.