On the product system of a completely positive semigroup

Given a W∗-continuous semigroup φ of unital, normal, completely positive maps of B(H), we introduce its continuous tensor product system . If α is a minimal dilation E₀-semigroup of φ with Arveson product system F, then is canonically isomorphic to F. We apply this construction to a class of semigroups of arising from a modified Weyl-Moyal quantization of convolution semigroups of Borel probability measures on . This class includes the heat flow on the CCR algebra studied recently by Arveson. We prove that the minimal dilations of all such semigroups are completely spatial, and additionally, we prove that the minimal dilation of the heat flow is cocyle conjugate to the CAR/CCR flow of index two.     

A short proof of the 3d distance theorem

We give a simple proof of the 3d distance conjecture.     

A note on the power semigroup of a completely simple semigroup

We prove the pseudovariety generated by power semigroups of completely simple semigroups is the semidirect product of the pseudovariety of block groups with the pseudovariety of right zero semigroups, and hence is decidable. This answers a question of Almeida from over 15 years ago. The author was supported in part by NSERC.     

An alternative proof of a Radon-Nikodym theorem for lattice homomorphisms

We give a new proof of the Luxemburg-Schep theorem for lattice homomorphisms.     

On the reduction of a contraction semigroup to a completely non unitary semigroup

According to a Theorem of B. Sz.-Nagy and C. Foia?, every strongly continuous semigroup of contraction operators on a Hilbert space, can be decomposed into a completely non unitary part and a unitary part. In this note we wish to show that by appropriately perturbing its generator, a contraction semigroup can be reduced to a completely non unitary one. In control theory, such a perturbation is related to the so called state feedback, and the reduction presented here has application in the problem of stabilizing linear control systems on a Hilbert space. This will be briefly discussed.     

A semigroup proof of a non-ergodic Markov process central limit theorem with third order terms

A semigroup product for linear operators is used to get a central limit theorem with third order terms which applies to all ergodic and non-ergodic discrete time Markov processes with continuous transition probability densities on a compact Hausdorff space.     

An alternative proof of the mountain pass theorem for a class of functionals

We present an alternative proof of the Mountain Pass Theorem by means of the classical Ekeland Variational Principle for a class of ${\mathcal{C}^1}$ -functionals. In this new proof we avoid the machinery of convex analysis by a simpler characterization of the critical values of the functional.     

A proof of the montague-plemmons-schein theorem on maximal subgroups of the semigroup of binary relations

The theorem in question is that the group of automorphisms of a partially ordered set (X,π), π denoting the order relation on the set X, is isomorphic to the maximal subgroup of ℬ_x containing π, where ℬ_x is the semigroup of all binary relations on X. This theorem is due to Montague and Plemmons [1] for the case X finite or countably infinite, and was extended by Schein to the general case, using a theorem due to Zaretsky [4]. A proof of the general case, based on [1] and results due to Plemmons and West [3], is also given in the preceding note by Plemmons and Schein [2]. The purpose of this note is to give an entirely self-contained proof of this intersesting theorem.     

Structure of continuous functions of a completely regular space

It is proved that any completely regular topological space is determined up to homeomorphism by the topological lattice (with the topology of pointwise convergence) of all its continuous real-valued functions. The well-known result of Kaplansky (for compact spaces) is a corollary of this theorem.Translated from Matematicheskie Zametki, Vol. 19, No. 6, pp. 863–869, June, 1976.     

On the maximalC *-algebra of zeros of completely positive mapping and on the boundary of a dynamic semigroup

Mathematical Notes -     

A stable theorem of the alternative: an extension of the gordan theorem

A theorem with a number of equivalent alternatives is proposed as an extension of the classical Gordan theorem of the alternative. The theorem can handle nonzero unrestricted variables which cannot be directly treated by ordinary theorems of the alternative. Like the Gordan theorem, the extended theorem has the stability feature that small perturbations in the data will not invalidate an alternative that is in force. The theorem has useful applications in establishing the boundedness and uniqueness of feasible points of polyhedral sets and of solutions to linear programming problems.     

A generalisation of a theorem of Koldunov with an elementary proof

We generalize a Theorem of Koldunov [2] and prove that a disjointness proserving quasi-linear operator between Resz spaces has the Hammerstein property.     

A simple proof of the ergodic theorem using nonstandard analysis

A simple proof of the individual ergodic theorem is given. The essential tool is the nonstandard measure theory developed by P. Loeb. Any dynamical system on an abstract Lebesgue space can be represented as a factor of a “cyclic” system with a hyperfinite cycle. The ergodic theorem for such a “cyclic” system is almost trivial because of its simple structure. The general case follows from this special case.     

A proof of the butterfly theorem by an argument from statics

The butterfly theorem is proved by assigning point masses to the four vertices of the wings and using the distributive property of the mass centre of a mechanical system.