2,810 research outputs found
A Common Framework for Restriction Semigroups and Regular *-Semigroups
Left restriction semigroups have appeared at the convergence of several flows of research, including the theories of abstract semigroups, of partial mappings, of closure operations and even in logic. For instance, they model unary semigroups of partial mappings on a set, where the unary operation takes a map to the identity map on its domain. This perspective leads naturally to dual and two-sided versions of the restriction property. From a varietal perspective, these classes of semigroupsâmore generally, the corresponding classes of Ehresmann semigroupsâderive from reducts of inverse semigroups, now taking a to a+=aaâ1 (or, dually, to aâ=aâ1a, or in the two-sided version, to both). In this paper the notion of restriction semigroup is generalized to P-restriction semigroup, derived instead from reducts of regular â-semigroups (semigroups with a regular involution). Similarly, [left, right] Ehresmann semigroups are generalized to [left, right] P-Ehresmann semigroups. The first main theorem is an abstract characterization of the posets P of projections of each type of such semigroup as âprojection algebrasâ. The second main theorem, at least in the two-sided case, is that for every P-restriction semigroup S there is a P-separating representation into a regular â-semigroup, namely the âMunnâ semigroup on its projection algebra, consisting of the isomorphisms between the algebraâs principal ideals under a modified composition. This theorem specializes to known results for restriction semigroups and for regular â-semigroups. A consequence of this representation is that projection algebras also characterize the posets of projections of regular â-semigroups. By further characterizing the sets of projections âinternallyâ, we connect our universal algebraic approach with the classical approach of the so-called âYork schoolâ. The representation theorem will be used in a sequel to show how the structure of the free members in some natural varieties of (P-)restriction semigroups may easily be deduced from the known structure of associated free inverse semigroups
E-semigroups Subordinate to CCR Flows
The subordinate E-semigroups of a fixed E-semigroup are in one-to-one
correspondence with local projection-valued cocycles of that semigroup. For the
CCR flow we characterise these cocycles in terms of their stochastic
generators, that is, in terms of the coefficient driving the quantum stochastic
differential equation of Hudson-Parthasarathy type that such cocycles
necessarily satisfy. In addition various equivalence relations and order-type
relations on E-semigroups are considered, and shown to work especially well in
the case of those semigroups subordinate to the CCR flows by exploiting our
characterisation.Comment: 14 pages; to appear in Communications on Stochastic Analysis. Minor
modifications made from version
Quantum Feynman-Kac perturbations
We develop fully noncommutative Feynman-Kac formulae by employing quantum
stochastic processes. To this end we establish some theory for perturbing
quantum stochastic flows on von Neumann algebras by multiplier cocycles.
Multiplier cocycles are constructed via quantum stochastic differential
equations whose coefficients are driven by the flow. The resulting class of
cocycles is characterised under alternative assumptions of separability or
Markov regularity. Our results generalise those obtained using classical
Brownian motion on the one hand, and results for unitarily implemented flows on
the other.Comment: 27 pages. Minor corrections to version 2. To appear in the Journal of
the London Mathematical Societ
How to differentiate a quantum stochastic cocycle.
Two new approaches to the infinitesimal characterisation of quantum stochastic cocycles are reviewed. The first concerns mapping cocycles on an operator space and demonstrates the role of H\"older continuity; the second concerns contraction operator cocycles on a Hilbert space and shows how holomorphic assumptions yield cocycles enjoying an infinitesimal characterisation which goes beyond the scope of quantum stochastic differential equations
- âŚ