Operator algebra

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

Fusion algebra at a rational level and cohomology of nilpotent subalgebras of\(\widehat{\mathfrak{s}\mathfrak{l}}\)(2)

Abstract: We define and calculate the fusion algebra of a WZW model at a rational level using cohomological methods. As a byproduct, we obtain a cohomological characterization of admissible representations of\(\widehat{\mathfrak{s}\mathfrak{l}}\)2.

Algebraic Aspects in Modular Theory

Abstract: Nowadays, the so called modular theory for von Neumann algebras is well-established and fully utilized (see [17] for example) in the field of operator algebras. The symbols conventionally used there, however, does not seem to be so much expressive in some sense. One of the main purposes in the present article is a focussed account of the problem of this kind for modular theory in operator algebras. In the past time, there had been already some suggestions on the improvement of notations for the modular theory but not in a thorough way. Among them, Woronowicz's approach [22] and new symbols introduced in [2] are worthy of attention. Around the same time of these works, the non-commutative //-theory for arbitrary von Neumann algebras came out and had been developped by several people such as Haagerup, Connes-Hilsum, Kosaki, and Araki-Masuda, and so on. It is worth pointing out the fact that, in this theory, the l/p-th power of a state of a von Neumann algebra is identified with an element in the relevant //-space. Now we give a brief outline of the contents in this article. The first section surveys the background materials and the substantial parts start from the next section.

On the existence of quantum subdynamics

Abstract: It is shown, using only elementary operator algebra, that an open quantum system coupled to its environment will have a subdynamics (reduced dynamics) as an exact consequence of the reversible dynamics of the composite system only when the states of system and environment are uncorrelated. Furthermore, it is proved that for a finite temperature the KMS condition for the lowest-order correlation function cannot be reproduced by any type of linear subdynamics except the reversible Hamiltonian one of a closed system. The first statement can be seen as a particular case of a more general theorem of Takesaki on the properties of conditional expectations in von Neumann algebras. The concept of subdynamics used here allows for memory effects, no assumption is made of a Markov property. For dynamical systems based on commutative algebras of observables the subdynamics always exists as a stochastic process in the random variable defining the open subsystem.

Crossed products of $L^p$ operator algebras and the K-theory of Cuntz algebras on $L^p$ spaces

Abstract: For $p \in [1, \infty),$ we define and study full and reduced crossed products of algebras of operators on $\sigma$-finite $L^p$ spaces by isometric actions of second countable locally compact groups. We give universal properties for both crossed products. When the group is abelian, we prove the existence of a dual action on the full and reduced $L^p$ operator crossed products. When the group is discrete, we construct a conditional expectation to the original algebra which is faithful in a suitable sense. For a free action of a discrete group on a compact metric space $X,$ we identify all traces on the reduced $L^p$ operator crossed product, and if the action is also minimal we show that the reduced $L^p$ operator crossed product is simple. We prove that the full and reduced $L^p$ operator crossed products of an amenable $L^p$ operator algebra by a discrete amenable group are again amenable. We prove a Pimsner-Voiculescu exact sequence for the K-theory of reduced $L^p$ operator crossed products by ${\mathbb{Z}}.$ We show that the $L^p$ analogs ${\mathcal{O}}_d^p$ of the Cuntz algebras ${\mathcal{O}}_d$ are stably isomorphic to reduced $L^p$ operator crossed products of stabilized $L^p$ UHF algebra by ${\mathbb{Z}},$ and show that $K_0 ({\mathcal{O}}_d^p) \cong {\mathbb{Z}} / (d - 1) {\mathbb{Z}}$ and $K_1 ({\mathcal{O}}_d^p) = 0.$

A comment on a recent work of Borchers

Abstract: We establish a converse of a recent result of Borchers