Operator algebra

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

Hereditary subalgebras of operator algebras

Abstract: In recent work of the second author, a technical result was proved establishing a bijective correspondence between certain open projections in a C*-algebra containing an operator algebra A, and certain one-sided ideals of A. Here we give several remarkable consequences of this result. These include a generalization of the theory of hereditary subalgebras of a C*-algebra, and the solution of a ten year old problem on the Morita equivalence of operator algebras. In particular, the latter gives a very clean generalization of the notion of Hilbert C*-modules to nonselfadjoint algebras. We show that an `ideal' of a general operator space X is the intersection of X with an `ideal' in any containing C*-algebra or C*-module. Finally, we discuss the noncommutative variant of the classical theory of `peak sets'.

The free entropy dimension of hyperfinite von Neumann algebras

Abstract: Suppose M is a hyperfinite von Neumann algebra with a normal, tracial state φ and {α 1 ,...,α n } is a set of selfadjoint generators for M. We calculate δ 0 (α 1 ,...,α n ), the modified free entropy dimension of {α 1 ,...,α n }. Moreover, we show that δ 0 (α 1 ,...,α n ) depends only on M and Φ. Consequently, δ 0 (α 1 ,...,α n ) is independent of the choice of generators for M. In the course of the argument we show that if {b 1 ,...,b n } is a set of selfadjoint generators for a von Neumann algebra R with a normal, tracial state and {b 1 ,...,b n } has finite-dimensional approximants, then δ 0 (N) < δ 0 (b 1 ,...,b n ) for any hyperfinite von Neumann subalgebra N of R. Combined with a result by Voiculescu, this implies that if R has a regular diffuse hyperfinite von Neumann subalgebra, then δ 0 (b 1 ,....,b n ) = 1.

Intrinsic non-commutativity of closed string theory

Abstract: We show that the proper interpretation of the cocycle operators appearing in the physical vertex operators of compactified strings is that the closed string target is noncommutative. We track down the appearance of this non-commutativity to the Polyakov action of the flat closed string in the presence of translational monodromies (i.e., windings). In view of the unexpected nature of this result, we present detailed calculations from a variety of points of view, including a careful understanding of the consequences of mutual locality in the vertex operator algebra, as well as a detailed analysis of the symplectic structure of the Polyakov string. We also underscore why this non-commutativity was not emphasized previously in the existing literature. This non-commutativity can be thought of as a central extension of the zero-mode operator algebra, an effect set by the string length scale — it is present even in trivial backgrounds. Clearly, this result indicates that the α ′ → 0 limit is more subtle than usually assumed.

New insight into the Witten-Dijkgraff-Verlinde-Verlinde equation

Abstract: We show that Witten-Dijkgraaf-Verlinde-Verlinde equation underlies the construction of N=4 superconformal multi--particle mechanics in one dimension, including a N=4 superconformal Calogero model.