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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 general theory of unitary representations is developed for these objects as harmonic analysis, which provides a good theoretical framework for the detailed study of the unitary representation of the classical Lie groups, and is regarded as an extension of the Fourier analysis to a general context.
Abstract: Classical Lie groups are important examples in the category of locally compact groups. The general theory of unitary representations is developed for these objects as harmonic analysis, which provides us a good theoretical framework for the detailed study of the unitary representations of the classical Lie groups. This is regarded as an extension of the Fourier analysis to a general context. For a locally compact group, its dual i.e. the set of all the equivalence classes of irreducible unitary representations plays an important role, and the duality established by Pontrjagin for Abelian groups, Tannaka and Krein for compact groups, Steinspring for unimodular groups, Eymard and Tatsuuma for locally compact groups is an important theoretical basis for the harmonic analysis. On the other hand, at the formal level in the framework of pure algebras, we use the notion of Hopf algebras to deal with the algebraic groups, discrete groups, or the dual of those objects at the same time. Then functional analysis is necessarily combined with the algebraic framework of Hopf algebras to have a good control with the infinite dimensional unitary representations. This theory, especially the argument utilized by Steinspring, suggests us to introduce the notion of Kac algebras in the language of von Neumann algebras. The first take off from the group or the group algebra to the Kac algebra was considered by Kac [7] and performed by Takesaki [23] by introducing the, so-called, Kac-Takesaki operator or the fundamental operator for the semifinite i.e. the unimodular case, and then completed by Enock and Schwartz [4, 20, 5] for the general case, in which the above mentioned duality was established by Takesaki, Enock and Schwartz, and others [24, 21].

100 citations

Journal ArticleDOI
TL;DR: In this paper, the ideal structure of C*-algebras arising from C*correspondences was studied and it was shown that the gauge-invariant ideals of C *-algeses are parameterized by certain pairs of ideals of original C*alges.
Abstract: We study the ideal structure of C*-algebras arising from C*-correspondences. We prove that gauge-invariant ideals of our C*-algebras are parameterized by certain pairs of ideals of original C*-algebras. We show that our C*-algebras have a nice property that should be possessed by a generalization of crossed products. Applications to crossed products by Hilbert C*-bimodules and relative Cuntz?Pimsner algebras are also discussed.

100 citations

Journal ArticleDOI
TL;DR: In this paper, the authors provide a rigorous mathematical foundation for strongly rational, holomorphic vertex operator algebras V of central charge c = 8, 16 and 24 initiated by Schellekens.
Abstract: We provide a rigorous mathematical foundation to the study of strongly rational, holomorphic vertex operator algebras V of central charge c = 8, 16 and 24 initiated by Schellekens. If c = 8 or 16 we show that V is isomorphic to a lattice theory corresponding to a rank c even, self-dual lattice. If c = 24 we prove, among other things, that either V is isomorphic to a lattice theory corresponding to a Niemeier lattice or the Leech lattice, or else the Lie algebra on the weight one subspace V 1 is semisimple (possibly 0) of Lie rank less than 24.

100 citations

Journal ArticleDOI
TL;DR: In this paper, a two-dimensional topological sigma model on a generalized Calabi-Yau target space was constructed in Batalin-Vilkovisky formalism using only a generalized complex structure and a pure spinor on the target space.
Abstract: A two-dimensional topological sigma-model on a generalized Calabi-Yau target space $X$ is defined. The model is constructed in Batalin-Vilkovisky formalism using only a generalized complex structure $J$ and a pure spinor $\rho$ on $X$. In the present construction the algebra of $Q$-transformations automatically closes off-shell, the model transparently depends only on $J$, the algebra of observables and correlation functions for topologically trivial maps in genus zero are easily defined. The extended moduli space appears naturally. The familiar action of the twisted N=2 CFT can be recovered after a gauge fixing. In the open case, we consider an example of generalized deformation of complex structure by a holomorphic Poisson bivector $\beta$ and recover holomorphic noncommutative Kontsevich $*$-product.

100 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that a σ-unital C*-algebra admits a countable approximate unit, i.e., it is stable, if and only if for each positive elementa∈A and eache>0 there exists a positive elementb∈Asuch that ‖ab‖

100 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202337
202277
2021125
2020141
2019173
2018169