A new representation theorem for completely [O-] simple semigroups

A semigroup is completely [O-] simple iff it is isomorphic to a semigroup ϕ of binary Boolean (I×I)-matrices (where I is a set) such that for every p,q∈I there exists precisely one matrix (a_ij)∈ϕ such that a_pg=1. Every such isomorphic representation of a completely [O-] simple semigroup is an arbitrary inflation of a reduced representation and all reduced representations are equivalent. Dedicated to Lazar Matveevič Gluskin on the occasion of his fiftieth birthday.     

A class of unary semigroups admitting a Rees matrix representation

In Billhardt et al. (Semigroup Forum 79:101–118, 2009) the authors introduced the notion of an associate inverse subsemigroup of a regular semigroup, extending the concept of an associate subgroup of a regular semigroup, first presented in Blyth et al. (Glasgow Math. J. 36:163–171, 1994). The main result of the present paper, Theorem 2.15, establishes that a regular semigroup S with an associate inverse subsemigroup S ^? satisfies three simple identities if and only if it is isomorphic to a generalised Rees matrix semigroup M(T;A,B;P), where T is a Clifford semigroup, A and B are bands, with common associate inverse subsemigroup E(T) satisfying the referred identities, and P is a sandwich matrix satisfying some natural conditions. If T is a group and A, B are left and right zero semigroups, respectively, then the structure described provides a usual Rees matrix semigroup with normalised sandwich matrix, thus generalising the Rees matrix representation for completely simple semigroups.     

Commutative semigroups with cancellation law: a representation theorem

Any commutative, cancellative semigroup S with 0 equipped with a uniformity can be embedded in a topological group \(\widetilde{S}\). We introduce the notion of semigroup symmetry T which enables us to turn \(\widetilde{S}\) into an involutive group. In Theorem 2.8 we prove that if S is 2-torsion-free and T is 2-divisible then the decomposition of elements of \(\widetilde{S}\) into a sum of elements of the symmetric subgroup \(\widetilde{S}_{s}\) and the asymmetric subgroup \(\widetilde{S}_{a}\) is polar. In Theorem 3.7 we give conditions under which a topological group \(\widetilde{S}\) is a topological direct sum of its symmetric subgroup \(\widetilde{S}_{s}\) and its asymmetric subgroup \(\widetilde{S}_{a}\). Theorem 2.8 and Theorem 3.7 are designed to be useful tools in studying Minkowski–Rådström–Hörmander spaces (and related topological groups \(\widetilde{S}\)), which are natural extensions of semigroups of bounded closed convex subsets of real Hausdorff topological vector spaces.     

A Hille-Yosida theorem for Bi-continuous semigroups

In order to treat one-parameter semigroups of linear operators on Banach spaces which are not strongly continuous, we introduce the concept of bi-continuous semigroups defined on Banach spaces with an additional locally convex topology . On such spaces we define bi-continuous semigroups as semigroups consisting of bounded linear operators which are locally bi-equicontinuous for and such that the orbit maps are -continuous. We then apply the result to semigroups induced by flows on a metric space as studied by J. R. Dorroh and J. W. Neuberger.     

A new factor theorem for absolute matrix summability

Abstract

In this paper, we have extended a known theorem dealing with |, p_n|k summability factors to the |A, p_n|k summability under weaker conditions by using an almost increasing sequence     

Goldie's theorem for semigroups

In this paper, we try to look for a semigroup analogue of Goldie's theorem which characterizes the orders of the full matrix ringM _n (D) over a division ringD. The main result of this paper, Theorem 4.11, does give a characterization of the semigroups which have a classical left quotient semigroupM _n (G ⁰), whereG ⁰ is a group with zero. The author is grateful to the referee for his careful modification of this paper and helpful comments which resulted in several improvements.     

A Wiener-Young type theorem for dual semigroups

The purpose of this paper is to obtain extensions of the Wiener-Young theorem for strongly continuous semigroups of positive operators in Banach lattices.     

A Tauberian theorem for strong Feller semigroups

We prove that a weakly ergodic, eventually strong Feller semigroup on the space of measures on a Polish space converges strongly to a projection onto its fixed space.     

A Hille-Yosida theorem for weakly continuous semigroups

We introduce a new class of weakly continuous semigroups and give a characterization of their infinitesimal generators, generalizing the classical Hille-Yosida Theorem for strongly continuous semigroups. The results are illustrated by the example of transition semigroups corresponding to the solutions of certain stochastic differential equations.     

A quantum analogue of Hunt's representation theorem for the generator of convolution semigroups on Lie groups

Summary In quantum mechanics certain operator-valued measures are introduced, called instruments, which are an analogue of the probability measures of classical probability theory. As in the classical case, it is interesting to study convolution semigroups of instruments on groups and the associated semigroups of probability operators. In this paper the case is considered of a finite-dimensional Hilbert space (n-level quantum system) and of instruments defined on a finite-dimensional Lie group. Then, the generator of a continuous semigroup of (quantum) probability operators is characterized. In this way a quantum analogue of Hunt's representation theorem for the generator of convolution semigroups on Lie groups is obtained.     

A representation theorem for quantum systems

In this note representations of quantum systems are investigated. We propose a unital bipolar theorem for unital quantum cones, which plays a key role in proving a representation theorem for quantum systems. It turns out that each quantum system is identified with a certain quantum L^∞-system up to a quantum order isomorphism.     

A complete representation theorem for G-martingales

In this article, we establish a complete representation theorem for G-martingales. Unlike the existing results in the literature, we provide the existence and uniqueness of the second-order term, which corresponds to the second-order derivative in Markovian case. The main ingredient of the article is a new norm for that second-order term, which is based on an operator introduced by Song.