Operator algebra

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

Derivations on the algebra of measurable operators affiliated with a type I von Neumann algebra

Abstract: Let M be a type I von Neumann algebra with the center Z and let LS(M) be the algebra of all locally measurable operators affiliated with M. We prove that every Z-linear derivation on LS(M) is inner. In particular, all Z-linear derivations on the algebras of measurable and respectively totally measurable operators are spatial and implemented by elements of LS(M).

Max-Plus Algebra and Applications to System Theory and Optimal Control

Abstract: In the modeling of human activities, in contrast to natural phenomena, quite frequently only the operations max (or min) and + are needed. A typical example is the performance evaluation of synchronized processes such as those encountered in manufacturing (dynamic systems made up of storage and queuing networks). Another typical example is the computation of a path of maximum weight in a graph and more generally of the optimal control of dynamical systems. We give examples of such situations. The max-plus algebra is a mathematical framework well suited to handle such situations. We present results on (i) linear algebra, (ii) system theory, and (iii) duality between probability and optimization based on this algebra.

On One-Parameter Subgroups of *-Automorphisms on Operator Algebras and the Corresponding Unbounded Derivations

Abstract: 1. It is known that if a derivation in C*-algebras is everywhere defined, then it is bounded and is weakly inner ([8]). On the other hand, in mathematical physics we have to confront unbounded derivations which are defined as infinitesimal generators of oneparameter subgroups of *-automorphisms on C*-algebras. Under some assumptions (for example, the positivity of infinitesimal operators), we may reduce the study of those unbounded derivations to the one of bounded derivations ([9]). However there are many important derivations in mathematical physics which do not satisfy the positivity. In this paper we wish to initiate a study of unbounded derivations in C*-algebras. A main goal of the study is to show that unbounded derivations in some C*-algebras, which are important in mathematical physics can be approximated by inner derivations on them. Obviously we can not expect such results for an arbitrary C*-algebra. For example, if 9f = C0(cc, xc) is the C*-algebra of all continuous functions on (ox, xo), vanishing at infinity, then the only bounded derivation is the identically zero mapping. On the other hand, let 8 be the differential operator d/ dt corresponding to the one-parameter subgroup of translation operators; then it is an unbounded derivation, so that it can not be approximated by the identically zero mapping in any reasonable sense. Now we shall explain briefly the result obtained in this paper. Let W be a uniformly hyperfinite C*-algebra, (p(t)I ox p(t)(a) is norm-continuous for each a E X, and let 8 be the infinitesimal generator corresponding to

Loop representations for (2+1) gravity on a torus

Abstract: We study the loop representation of the quantum theory for 2+1-dimensional general relativity on a manifold M=T2*R, where T2 is the torus, and compare it with the connection representation for this system. In particular, we look at the loop transform in the part of the phase space where the holonomies are boosts, and study its kernel. This kernel is dense in the connection representation, and the transform is not continuous with respect to the natural topologies, even in its domain of definition. Nonetheless, loop representations isomorphic to the connection representation corresponding to this part of the phase space can still be constructed if due care is taken. We present this construction, but note that certain ambiguities remain; in particular, functions of loops cannot be uniquely associated with functions of connections.