arXiv subjects
Publications and source records attributed to J. D. Maitland Wright.
Let A be a C*-algebra. It is shown that A is an AW*-algebra if, and only if, each maximal abelian self--adjoint subalgebra of A is monotone complete. An analogous result is proved for Rickart C*-algebras; a C*-algebra is a Rickart C*-algebra if, and only if, it is unital and each maximal abelian self--adjoint subalgebra of A is monotone σ-complete.
Let S be the Stone space of a complete, non-atomic Boolean algebra. Let G be a countably infinite group of homeomorphisms of S. Let the action of G on S have a free dense orbit. Then we prove that, on a generic subset of S, the orbit equivalence relation coming from this action can also be obtained as an action of the (abelian) Dyadic Group. For the special case where the complete Boolean algebra is the algebra of regular open subsets of the real numbers, this reduces to a theorem of Sullivan-Weiss-Wright. By applying our new dynamical results we improve on an earlier paper by constructing a family of monotone complete C*-algebras, {B(t):t in T} with the following properties. First,T is large; it can be identified with the set of all subsets of the reals. Secondly each B(t) is a small C*-algebra,which is a monotone complete factor of Type III, is hyperfinite and is not a von Neumann algebra.Thirdly, when t is not equal to s, then B(s) is not isomorphic to B(t). In fact, B(s) and B(t) take different values in the classification semi-group for small monotone complete C*-algebras.
A general `quantum history theory' can be characterised 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 $H \otimes H$, 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 $H \otimes H$.
Let $A$ be a von Neumann algebra with no direct summand of Type $\roman I_2$, and let $\scr P(A)$ be its lattice of projections. Let $X$ be a Banach space. Let $m\:\scr P(A)\to X$ be a bounded function such that $m(p+q)=m(p)+m(q)$ whenever $p$ and $q$ are orthogonal projections. The main theorem states that $m$ has a unique extension to a bounded linear operator from $A$ to $X$. In particular, each bounded complex-valued finitely additive quantum measure on $\scr P(A)$ has a unique extension to a bounded linear functional on $A$.