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Operator algebra
About: Operator algebra is a research topic. Over the lifetime, 5783 publications have been published within this topic receiving 165303 citations.
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TL;DR: In this paper, a generalization of vector-valued frame theory, called operator-valued frames, is presented, which can be viewed as the multiplicity-tiplicity-one case and extends to higher multiplicity their dilation approach.
Abstract: We develop a natural generalization of vector-valued frame theory, we term operator-valued frame theory, using operator-algebraic methods. This extends work of the second author and D. Han which can be viewed as the mul- tiplicity one case and extends to higher multiplicity their dilation approach. We prove several results for operator-valued frames concerning duality, dis- jointeness, complementarity , and composition of operator valued frames and the relationship between the two types of similarity (left and right) of such frames. A key technical tool is the parametrization of Parseval operator val- ued frames in terms of a class of partial isometries in the Hilbert space of the analysis operator. We apply these notions to an analysis of multiframe gener- ators for the action of a discrete group G on a Hilbert space. One of the main results of the Han-Larson work was the parametrization of the Parseval frame generators in terms of the unitary operators in the von Neumann algebra gen- erated by the group representation, and the resulting norm path-connectedness of the set of frame generators due to the connectedness of the group of unitary operators of an arbitrary von Neumann algebra. In this paper we general- ize this multiplicity one result to operator-valued frames. However, both the parameterization and the proof of norm path-connectedness turn out to be necessarily more complicated, and this is at least in part the rationale for this paper. Our parameterization involves a class of partial isometries of a different von Neumann algebra. These partial isometries are not path-connected in the norm topology, but only in the strong operator topology. We prove that the set of operator frame generators is norm pathwise-connected precisely when the von Neumann algebra generated by the right representation of the group has no minimal projections. As in the multiplicity one theory there are analogous results for general (non-Parseval) frames.
67 citations
TL;DR: In this paper, the authors give a complete description of all derivations on the above algebras of operators in the case of type I von Neumann algebra M. In particular, if M is of type i ∞ then every derivation on LS(M) (resp. S(M,τ) ) is inner.
67 citations
TL;DR: In this article, the Jacobi identity for intertwining operator algebras is introduced and the main properties of genus-zero conformal field theories, including vertex operator algebra, modules, intertwining operators, Verlinde algebra, and fusing and braiding matrices, are incorporated into this identity.
Abstract: We prove a generalized rationality property and a new identity that we call the ''Jacobi identity'' for intertwining operator algebras. Most of the main properties of genus-zero conformal field theories, including the main properties of vertex operator algebras, modules, intertwining operators, Verlinde algebras, and fusing and braiding matrices, are incorporated into this identity. Together with associativity and commutativity for intertwining operators proved by the author in [H4] and [H6], the results of the present paper solve completely the problem of finding a natural purely algebraic structure on the direct sum of all inequivalent irreducible modules for a suitable vertex operator algebra. Two equivalent definitions of intertwining operator algebra in terms of this Jacobi identity are given.
67 citations
TL;DR: In this paper, the evolution of correlation functions of local fields in a two-dimensional quantum field theory under the λ T √ T$ deformation, suitably regularized, was studied.
Abstract: We study the evolution of correlation functions of local fields in a two-dimensional quantum field theory under the $\lambda T\bar T$ deformation, suitably regularized. We show that this may be viewed in terms of the evolution of each field, with a Dirac-like string being attached at each infinitesimal step. The deformation then acts as a derivation on the whole operator algebra, satisfying the Leibniz rule. We derive an explicit equation which allows for the analysis of UV divergences, which may be absorbed into a non-local field renormalization to give correlation functions which are UV finite to all orders, satisfying a (deformed) operator product expansion and a Callan-Symanzik equation. We solve this in the case of a deformed CFT, showing that the Fourier-transformed renormalized two-point functions behave as $k^{2\Delta+2\lambda k^2}$, where $\Delta$ is their IR conformal dimension. We discuss in detail deformed Noether currents, including the energy-momentum tensor, and show that, although they also become non-local, when suitably improved they remain finite, conserved and satisfy the expected Ward identities. Finally, we discuss how the equivalence of the $T\bar T$ deformation to a state-dependent coordinate transformation emerges in this picture.
67 citations
TL;DR: A relatively simple definition of a locally compact quantum group in the C*-algebra setting will be explained as it was recently obtained by the authors.
Abstract: A relatively simple definition of a locally compact quantum group in the C*-algebra setting will be explained as it was recently obtained by the authors. At the same time, we put this definition in the historical and mathematical context of locally compact groups, compact quantum groups, Kac algebras, multiplicative unitaries, and duality theory.
67 citations