Generators for categories of compact irreducible semigroups

For a compact totally ordered space X, K. H. Hofmann and P. S. Mostert constructed a topological semigroup Irr (X)₀ such that every compact irreducible semigroup with idempotents X is a surmorphic image of Irr (X)₀ [2]. J. H. Carruth and M. Mislove pointed out that Irr (X)₀ was in general not a compact semigroup. In this paper a compact connected hormos will be constructed which contains Irr (X)₀ as a dense subsemigroup and for which every compact irreducible semigroup with idempotents X is a surmorphic image. This leads to a new proof of the existence of generators for the category of compact irreducible semigroups with idempotents X. It will then be shown that Irr (X)₀ contains generators for the category.     

Nonlocal Cauchy problem for delay integrodifferential equations of Sobolev type in Banach spaces

In this paper, we prove the existence of mild and strong solutions of nonlinear time varying delay integrodifferential equations of Sobolev type with nonlocal conditions in Banach spaces. The results are obtained by using the theory of compact semigroups and Schaefer's fixed-point theorem.     

Compact quantum groups and group duality

In this paper, we consider the *-representations of compact quantum groups and group duality. The main results in the paper are: (1) there is a one-to-one correspondence between the *-representations of compact quantum groups and *-representations of the dual Banach *-algebra; (2) the category of commutative compact quantum groups (semigroups) is a dual category to the category of compact groups (semigroups); (3) the dual category of the category of locally compact groups (semigroups) is the category of commutative Hopf C^*-algebras with a particular property. Our group duality has the flavor of a Gelfand-Naimark type theorem for compact quantum groups, and for Hopf C^*-algebras.     

Ω-compact semigroups having congruence extension property

The congruence extension property (CEP) of semigroups has been extensively studied by a number of authors. We call a compact semigroup S an Ω-compact semigroup if the set of all regular elements of S forms an ideal of S. In this note, we characterize the Ω-compact semigroup having (CEP). Our result extends a recent result obtained by X.J. Guo on the congruence extension property of strong Ω-compact semigroups which is a semigroup containing precisely one regular D-class.     

Guarded and Banded Semigroups

The variety of guarded semigroups consists of all (S,·, ˉ) where (S,·) is a semigroup and x ↦ \overline{x} is a unary operation subject to four additional equations relating it to multiplication. The semigroup Pfn(X) of all partial transformations on X is a guarded semigroup if x \overline{f} = x when xf is defined and is undefined otherwise. Every guarded semigroup is a subalgebra of Pfn(X) for some X. A covering theorem of McAlister type is obtained. Free guarded semigroups are constructed paralleling Scheiblich's construction of free inverse semigroups. The variety of banded semigroups has the same signature but different equations. There is a canonical forgetful functor from guarded semigroups to banded semigroups. A semigroup underlies a banded semigroup if and only if it is a split strong semilattice of right zero semigroups. Each banded semigroup S contains a canonical subsemilattice g_⋆(S). For any given semilattice L, a construction to synthesize the general banded semigroup S with g_⋆ ≅ L is obtained.     

紧交换半群上概率测度卷积序列的极限性质

本文用“部分群化”的方法研究紧交换半群上概率测度的极限性质.§3讨论 i.i.d 的情形,将紧群上的 Kawada-It(?)型结果以相应的形式建立到紧交换半群上.§4讨论独立非同分布的情形,建立了紧交换半群上的强 Kloss 收敛准则,它曾由苏联学者 Maksimov先后在有限群([1])与紧群上([2])得到.     

局部紧半群上概率测度组合收敛的某些结果

本文讨论局部紧半群上概率测度的组合收敛性,主要结果是利用局部群化的方法给出了概率测度组合收敛的一些结果.     

Existence of invariant measures of stochastic systems with delay in the highest order partial derivatives

In this note, we shall consider the existence of invariant measures for a class of infinite dimensional stochastic functional differential equations with delay whose driving semigroup is eventually norm continuous. The results obtained are applied to stochastic heat equations with distributed delays which appear in such terms having the highest order partial derivatives. In these systems, the associated driving semigroups are generally non eventually compact.     

Compact totally disconnected k-divisible semigroups

One of the early results [5] regarding divisibility in semigroups states that no finite non-degenerate group is divisible. A sequel to this (which in view of well-known results on compact semigroups is a generalization) is that a compact semigroup is divisible if and only if each component is a divisible subsemigroup [2]. Consequently, a finite semigroup is divisible if and only if it is an idempotent semigroup. However, it is of some interest to know which finite semigroups are k-divisible for a given positive integerk≥2. In this note we present a complete characterization of finitek-divisible semigroups, and use this along with a result of K. Numakura [8], to characterize compact totally disconnected k-divisible semigroups     

On local integrability and transience of convolution semigroups of measures

The analytic proof due to M. Itô of the Kesten-Ornstein transience criterion for continuous convolution semigroups of nonnegative contraction measures on a compactly generated Abelian locally compact group has been reworked and given a self-contained form. The new proof still relies on the existence of the equilibrium measure but dispenses with the complete maximum principle.     

Fixed point properties of semigroups of non-expansive mappings

In recent years, there have been considerable interests in the study of when a closed convex subset K of a Banach space has the fixed point property, i.e. whenever T is a non-expansive mapping from K into K, then K contains a fixed point for T. In this paper we shall study fixed point properties of semigroups of non-expansive mappings on weakly compact convex subsets of a Banach space (or, more generally, a locally convex space). By considering the classes of bicyclic semigroups we answer two open questions, one posted earlier by the first author in 1976 (Dalhousie) and the other posted by T. Mitchell in 1984 (Virginia). We also provide a characterization for the existence of a left invariant mean on the space of weakly almost periodic functions on separable semitopological semigroups in terms of fixed point property for non-expansive mappings related to another open problem raised by the first author in 1976.     

Finite Groups in Which Each Irreducible Character has at Most Two Zeros

Let G be a finite group, Irr(G) denotes the set of irreducible complex characters of G and gG the conjugacy class of G containing element g. A well-known theorem of Burnside([1,Theorem 3.15]) states that every nonlinear X ∈ Irr(G) has a zero on G, that is, an element x (or a conjugacy class xG) of G with X(x) = 0. So, if the number of zeros of character table is very small, we may expect, the structure of group is heavily restricted. For example, [2, Proposition 2.7] claimes that G is a Frobenius group with a complement of order 2 if each row in charcter table has at most one zero (its proof uses the classification of simple groups). In this note, we characterize the finite group G satisfying the following hypothesis:     

Quantum stochastic convolution cocycles III

Every Markov-regular quantum Lévy process on a multiplier C ^*-bialgebra is shown to be equivalent to one governed by a quantum stochastic differential equation, and the generating functionals of norm-continuous convolution semigroups on a multiplier C ^*-bialgebra are then completely characterised. These results are achieved by extending the theory of quantum Lévy processes on a compact quantum group, and more generally quantum stochastic convolution cocycles on a C ^*-bialgebra, to locally compact quantum groups and multiplier C ^*-bialgebras. Strict extension results obtained by Kustermans, together with automatic strictness properties developed here, are exploited to obtain existence and uniqueness for coalgebraic quantum stochastic differential equations in this setting. Then, working in the universal enveloping von Neumann bialgebra, we characterise the stochastic generators of Markov-regular, *-homomorphic (respectively completely positive and contractive), quantum stochastic convolution cocycles.     

Idempotent probabilities on semigroups

Summary In this paper, idempotent probability measures have been considered on semigroups which are locally compact or metric and satisfy: (*) A ^–1 B and Ax ^–1 are compact whenever A and B are so, for every x in the semigroup. Such semigroups are more general than compact semigroups which do admit of such measures. On such semigroups we can construct such measures by the usual process if there is a compact sub-semigroup. It is shown in this paper that if such a measure exists in such semigroups, then it must be such an extension measure. Some related results concerning the conditions (*) are also discussed here.     

Approximative compactness of the algebraic sum of sets

Let X be a group with an invariant metric, A and B nonempty subsets of X with B compact. It is proved that if A is an existence set [1] (approximatively compact [2]) then A + B and B + A are existence sets (approximatively compact). An example is given of a one-dimensional linear metric space (with an invariant metric) in which there exist an approximatively compact set A and an element v such that A + v is not an existence set.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 55–60, January, 1978.     

Amenability and nonlinear flows in dual Banach spaces

This paper is devoted to the study of nonlinear flows with weak* compact convex phase spaces sitting in conjugate Banach spaces. Nonlinear fixed point properties for the classes of amenable affine and convex semitopological semigroups are established; giving partial answers to question 1 in Lau and Zhang (J Funct Anal 263:2949–2977, 2012); we also prove related results for amenable semitopological semigroups. Furthermore, by almost periodicity techniques, we derive a non-commutative version of Bruck’s result (Pac J Math 53:59–70, 1974) on the existence of a nonexpansive retract; we also provide a characterization of left amenability property for the space of almost periodic functions for semitopological semigroups, and derive a fixed point property in \(\ell ^1\).     

Perturbation results for exponentially dichotomous operators on general Banach spaces

Some perturbation results for exponentially dichotomous operators are applied to prove the existence of stable and anti-stable solutions of Riccati equations associated to block operators on general Banach spaces, both for compact perturbations and for bisemigroups made up of immediately norm continuous semigroups.     

紧拓扑半群上概率测度卷积序列的极限性质

本文讨论紧拓扑半群上概率测度卷积序列的若干重要极限性质．在第１节中，我们讨论测度集的代数结构与其支撑集代数结构的关系．第２节的定理１，通过支撑集的代数结构给出组合收敛测度序列的一个极限定理．在定理２中我们讨论独立同分布时的情形，建立了一类紧半群上的Ｋａｗａｄａ－Ｉｔ型结果．这些定理推广了紧群、紧交换半群、紧Ｌ－Ｘ半群上一些相应的结论．     

Continuous state branching semigroups

Summary This paper is concerned with Markov processes with continuous creation where the phase space is a general separable compact metric space. The transition probabilities for such a process determine a semigroup of operators acting on a function space over the collection of bounded Borel measures on the phase space. Such a semigroup is characterized by a particular convolution condition and is called a continuous state branching semigroup. A connection is established between continuous state branching semigroups and certain semigroups of nonlinear operators and then this connection is exploited to establish an existence theorem for the former.Research associated with a project in probability at Princeton University supported by the Office of Army Research.