Mappings preserving Segal's entropy in von Neumann algebras

Abstract

We investigate the situation when a normal positive linear unital map on a semifinite von Neumann algebra leaving the trace invariant does not change the Segal entropy of the density of a normal, not necessarily normalised, state. Two cases are dealt with: a) no restriction on the map is imposed, b) the map represents a repeatable instrument in measurement theory which means that it is idempotent. Introduction In the paper, the question of invariance of Segal’s entropy under the action of a normal positive linear unital map is addressed in the case of a semifinite von Neumann algebra. The notion of Segal’s entropy was introduced by Segal in [9] for semifinite von Neumann algebras as a direct counterpart of von Neumann’s entropy defined for the full algebra B(H) of all bounded linear operators on a Hilbert space by means of the canonical trace. However, in the case of an arbitrary semifinite von Neumann algebra, where instead of the canonical trace we have a normal semifinite faithful trace, substantial differences between these two entropies arise. Perhaps the most fundamental one consists in the fact that while a normal state on B(H) is represented by a positive operator of trace one (the so-called ‘density matrix’), in the case of an arbitrary semifinite von Neumann algebra this ‘density matrix’ can be an unbounded operator. This prompted Segal to consider only the states whose ‘density matrices’ were in the algebra. In our analysis, we avoid this restriction as well as we allow the trace to be semifinite and not finite, the latter being also often assumed while dealing with Segal’s entropy. On the way to the main theorems, some auxiliary results about strict operator convexity or Jensen’s inequality for unbounded measurable operators are obtained which seem to be interesting and of some importance in their own right. 1. Preliminaries and notation Let M be a semifinite von Neumann algebra of operators acting on a Hilbert space H with a normal semifinite faithful trace τ , identity 1, and predual M∗. By M we shall denote the set of positive operators in M , and by M ∗ —the set of positive functionals in M∗. These functionals will be sometimes referred to as (nonnormalised) states. https://doi.org/10.5186/aasfm.2019.4439 2010 Mathematics Subject Classification: Primary 46L53; Secondary 81P45.