J. D. Maitland Wright

Generic dynamics and monotone complete C*-algebras

Soit R une relation dequivalence generique denombrable ergodique sur un espace parfait polonais X. Modulo un sous-ensemble maigre de X, R peut sidentifier a la relation dequivalence dorbite suivant une action canonique de Z

Quantum measures and states on Jordan algebras

A problem of Mackey for von Neumann algebras has been settled by the conjunction of the early work of Gleason and the recent advances of Christensen and Yeadon. We show that Mackeys conjecture holds in much greater generality. LetA be a JBW-algebra and letL be the lattice of all projections inA. A quantum measure onL is a countably additive map,m, fromL into the real numbers. Our results imply thatm always has a unique extension to a bounded linear functional onA, provided thatA has no TypeI2 direct summand.

C*-algebras which are Grothendieck spaces

Each monotoneσ-completeC*-algebra is a Grothendieck space.

Representing Yosida-Hewitt decompositions for classical and non-commutative vector measures

Abstract Let m be a bounded, real valued measure on a field of sets. Then, by the Yosida-Hewitt theorem, m has a unique decomposition into the sum of a countably additive and a singular measure. We show here that, in contrast to the classical arguments, this decomposition can be achieved by constructing the countably additive component. From this we obtain a simple formula for the countably additive part of a (strongly bounded) vector measure. We develop these ideas further by considering a weakly compact operator T on a von Neumann algebra M. It turns out that T has a unique decomposition into TN +TS, where TS is singular, TN is completely additive on projections and, for each x in M, there exists an increasing sequence of projections (pn)(n = 1,2…), such that T N ( x ) = lim u2061 T ( p n xp n ) . When M has a faithful representation on a separable Hilbert space, then we can fix a sequence of projections (pn)(n = 1,2…) such that the above equation holds for every choice of x in M. For general M, there exists an increasing net of projections such that, for every y in M, lim u2061 F ∥ T N ( y ) − T ( q F yq F ) ∥ = 0.

Operator algebras and a theorem of Dieudonne

LetA be aC*-algebra with second dualA″. Let (φn)(n=1,...) be a sequence in the dual ofA such that limφn(a) exists for eacha εA. In general, this does not imply that limφn(x) exists for eachx εA″. But if limφn(p) exists whenever p is the range projection of a positive self-adjoint element of the unit ball ofA, then it is shown that limφn(x) does exist for eachx inA″. This is a non-commutative generalisation of a celebrated theorem of Dieudonné. A new proof of Dieudonné’s theorem, for positive measures, is given here. The proof of the main result makes use of Dieudonné’s original theorem.

Homogeneous Decoherence Functionals¶in Standard and History Quantum Mechanics

Abstract: General history quantum theories are quantum theories without a globally defined notion of time. Decoherence functionals represent the states in the history approach and are defined as certain bivariate complex-valued functionals on the space of all histories. However, in practical situations – for instance in the history formulation of standard quantum mechanics – there often is a global time direction and the homogeneous decoherence functionals are specified by their values on the subspace of homogeneous histories.In this work we study the analytic properties of (i) the standard decoherence functional in the history version of standard quantum mechanics and (ii) homogeneous decoherence functionals in general history theories. We restrict ourselves to the situation where the space of histories is given by the lattice of projections on some Hilbert space ℋ. Among other things we prove the non-existence of a finitely valued extension for the standard decoherence functional to the space of all histories, derive a representation for the standard decoherence functional as an unbounded quadratic form with a natural representation on a Hilbert space and prove the existence of an Isham–Linden–Schreckenberg (ILS) type representation for the standard decoherence functional.

On tracial operator representations of quantum decoherence functionals

A general quantum history theory can be characterized by the space of histories and by the space of decoherence functionals. In this note we consider the situation where the space of histories is given by the lattice of projection operators on an infinite dimensional Hilbert space H. We study operator representations for decoherence functionals on this space of histories. We first give necessary and sufficient conditions for a decoherence functional being representable by a trace class operator on H⊗H, an infinite dimensional analogue of the Isham–Linden–Schreckenberg representation for finite dimensions. Since this excludes many decoherence functionals of physical interest, we then identify the large and physically important class of decoherence functionals which can be represented, canonically, by bounded operators on H⊗H.

Quantum logics and convex geometry

The main result is a representation theorem which shows that, for a large class of quantum logics, a quantum logic,Q, is isomorphic to the lattice of projective faces in a suitable convex setK. As an application we extend our earlier results [4], which, subject to countability conditions, gave a geometric characterization of those quantum logics which are isomorphic to the projection lattice of a von Neumann algebra or aJ B W-algebra.

Quasi completely continuous multilinear operators

We introduce the concept of quasi-completely continuous multilinear operators and use this concept to characterize, for a wide class of Banach spaces X1, …, Xk, the multilinear operators T : X1 × … × Xk → X with an X-valued Aron–Berner extension.

Functional Analysis for the Practical Man

Publisher Summary Practical analysis has its origins in geometry. All the theorems of classical analysis and positive results of elementary measure theory are theorems of ZF + DC. Of course, in functional analysis, extensive use has been made of the Axiom of Choice. But, even in functional analysis, many theorems are derived in ZF + DC—for example, the Closed Graph Theorem and the Uniform Boundedness Theorem. Of those theorems, such as the Hahn-Banach Theorem and the Krein-Milman Theorem, whose usual proofs depend on the Axiom of Choice, most can be derived in ZF + DC, provided some mild separability conditions are imposed.