Left Amenable Semigroup Research Articles

Density in Arbitrary Semigroups

We introduce some notions of density in an arbitrary semigroup S which extend the usual notions in countable left amenable semigroups in which density is based on Folner sequences. The new notions are based on nets of finite sets. We show that under certain conditions on the nets and on S these notions relate nicely to some established notions of size in S such as central, syndetic, and piecewise syndetic. And we investigate the conditions under which these notions have other desirable properties such as translation invariance. We obtain new information about the algebraic structure of the Stone-Cech compactification β S of S and derive generalizations of some known Ramsey Theoretic results, including Bergelson's density version of Schur's Theorem.

A counterexample to Sorenson’s conjecture: The finitely generated case

It is shown that there exists a finitely generated, left amenable semigroup whichis right cancellative but not left cancellative. This resolves negatively a questionraised by M. Klawe.

Existence of Invariant Weak Units in Banach Lattices: Countably Generated Left Amenable Semigroup of Operators

AbstractLet Σ be a countably generated left amenable semigroup and ﹛Tσ|σ ∈ Σ﹜ be a representation of Σ as a semigroup of positive linear operators on a weakly sequentially complete Banach lattice E with a weak unit e. It is assumed Tσ are uniformly bounded. It is shown that a necessary and sufficient condition for the existence of a weak unit invariant under ﹛Tσ | σ ∈ Σ﹜ is that inf σ∈Σ H(Tσe) > 0 for all nonzero H in the positive dual cone of E.

Fixed Point Theorem and Nonlinear Ergodic Theorem for Nonexpansive Semigroups Without Convexity

AbstractWe first prove a nonlinear ergodic theorem for nonexpansive semigroups without convexity in a Hilbert space. Further we prove a fixed point theorem for non-expansive semigroups without convexity which generalizes simultaneously fixed point theorems for left amenable semigroups and left reversible semigroups.

The positive fixed points of Banach lattices

Let Z Z be a Banach lattice endowed with positive cone C C and an order-continuous norm | | ⋅ | | || \cdot || . Let G G be a left-amenable semigroup of positive linear endomorphisms of Z Z . Then the positive fixed points C 0 {C_0} of Z Z under G G form a lattice cone, and their linear span Z 0 {Z_0} is a Banach lattice under an order-continuous norm | | ⋅ | | 0 || \cdot |{|_0} which agrees with | | ⋅ | | || \cdot || on C 0 {C_0} . A counterexample shows that under the given conditions Z 0 {Z_0} need not contain all the fixed points of Z Z under G G , and need not be a sublattice of ( Z , C ) (Z,C) . The paper concludes with a discussion of some related results.

The cardinality of the set of left invariant means on a left amenable semigroup

The cardinality of the set of left invariant means on a left amenable semigroup

On the dimension of left invariant means and left thick subsets

IfSis a left amenable semigroup, letdim⁡⟨Ml(S)⟩\dim \langle Ml(S)\rangledenote the dimension of the set of left invariant means onS. Theorem.If S is left amenable, thendim⁡⟨Ml(S)⟩=n>∞\dim \langle Ml(S)\rangle = n > \inftyif and only if S contains exactly n disjoint finite left ideal groups. This result was proved by Granirer forScountable or left cancellative. Moreover, whenSis infinite, left amenable, and either left or right cancellative, we show thatdim⁡⟨Ml(S)⟩\dim \langle Ml(S)\rangleis at least the cardinality ofS. An application of these results shows that the radical of the second conjugate algebra ofl1(S){l_1}(S)is infinite dimensional whenSis a left amenable semigroup which does not contain a finite ideal.

A characterization of topological left thick subsets in locally compact left amenable semigroups

A characterization of topological left thick subsets in locally compact left amenable semigroups

Amenability and invariant subspaces

Let E be a topological vector space (over the real or complex field). A well-known geometric form of the Hahn-Banach theorem asserts that if U is an open convex subset of E and M is a subspace of E which does not meet U, then there exists a closed hyperplane H containing M and not meeting U. In this paper we prove, among other things, that if S is a left amenable semigroup (which is the case, for example, when S is abelian or when S is a solvable group, see [3, p.11]), then for any right linear action of S on E, if U is an invariant open convex subset of E containing an invariant element and M is an invariant subspace not meeting U, then there exists a closed invariant hyperplane H of E containing M and not meeting U. Furthermore, this geometric property characterizes the class of left amenable semigroups.

Invariant normal positive functionals

Let (02, S, a) be a system consisting of a von Neumann algebra a, a semigroup S and an action 01: S x a+ @. We continue in this paper the study of the existence of invariant (invariant with respect to a) normal positive functionals on 6Y begun in [3]. We shall freely make use of the notations of [3]. In [3], we considered the case when 5’ was a left amenable semigroup and (Y. an antirepresentation of S as normal positive contractions on G?. In this paper we consider S to be amenable and 01 an antirepresentation of S as normal *homomorphisms on 0Z into GZ (i.e., for each s E S, 01~ is a normal *homomorphism and ast = ata,). For a given normal positive linear functional 9s on G?, we investigate conditions for the existence of an S-invariant normal positive linear functional y on a such that p,, < v (or q0 N v). As a consequence, we get conditions for G! to be S-finite.

The range of invariant means on locally compact groups and semigroups

This paper extends the results of Granirer and Chou concerning the range of a left invariant mean on a discrete semigroup to the case when S is any Borel subsemigroup of a locally compact group. 0. Introduction. Granirer has shown in [2] that for an infinite, right cancellation, left amenable discrete semigroup S, other than what he calls an AB group, there exists a nested family of left almost convergent subsets of S on which any left invariant mean (LIM) attains all values of the closed interval [0, 1]. That is, there exists a family {A(t)|t E [0, 1]} of subsets of S for which (i) s<t implies A(s)c A(t), and (ii) p(%A(t))=t for any LIM q on m(S). Chou, in [1], partially extended this result to the case when S is a group and obtained the following theorem thereby proving a conjecture made in [2]. THEOREM (CHOU). If S is an infinite right cancellation left amenable semigroup then the range of each LIM on m(S) is the whole [0, 1] interval. In this paper we extend these results to locally compact topological groups and obtain the following main theorems: THEOREM A. Let S be an infinite Borel subsemigroup of positive Haar measure in a locally compact group. If q is a LIM on L'(S) then there exists a nested family {A(t)lt E [0, 1 } of Borel subsets of S stuch that cp(A(t))=t for all t E [0, 1]. THEOREM B. Let S be an infinite Borel subsemigroup in a locally compact group and let A denote the algebra of bounded Borel measurable Received by the editors March 7, 1972. AMS (MOS) subject classifications (1969). Primary 2875, 2220; Secondary 4250, 4655. 1 The results in this paper constitute a portion of the author's Ph.D. thesis written under the direction of Dr. E. A. Granirer at the University of British Columbia. The author wishes to thank Professor Granirer for his aid and advice and as well the Canada Council for its financial support. ? American Mathematical Society 1973

Multipliers for the space of almost-convergent functions on a semigroup

Let S be a countably infinite left amenable cancellative semigroup, FL(S) the space of left almost-convergent functions on S. The purpose of this paper is to show that the following two statements concerning a bounded real function f on S are equivalent: (i) f -FL(S) c FL(S); (ii) there is a constant cx such that for each ->0 there exists a set A (S satisfying (a) p(XA)=0 for each left invariant mean 9 on S and (b) If(x)cx 0 and qp(lf)=qp(f) for seS and fenm(S), where l,fe m(S) is defined by (lsf)(t)=f(st), t E S. The set of left invariant means on S is denoted by ML(S). If ML(S) is nonempty, then S is said to be left amenable [2]. A bounded real function f on a left amenable semigroup is called left almost-convergent if (f) equals a fixed constant d(f) as 79 runs through ML(S) [2]. The set of all left almost-convergent functions, denoted by FL(S), is a vector subspace of m(S) and it contains constant functions. But, in general, it is not closed under multiplication. The purpose of this paper is to study this aspect of FL(S) and our main result is the following. THEOREM. Let S be a countable left-cancellative left amenable semigroup withoutfinite left ideals. Then the following two statements concerning a function f E m(S) are equivalent: (i) f is a multiplier of FL(S), i.e., f * FL(S) cFL(S); (ii) f is S-convergent to a constant x, i.e.,for a given e>0 there exists a set A c S such that (a) 79(XA)=0 for each 9p E ML(S), and (b) If(x)-xl <e if x E S\A. 2. From now on S will always denote a left-cancellative left amenable semigroup without finite left ideals. Presented to the Society, January 17, 1972; received by the editors May 31, 1972. AMS (MOS) subject class/ifcations (1970). Primary 43A07; Secondary 46N05.

Probability measures on semigroups

Let S S be a discrete semigroup, P P a probability measure on S S and s ∈ S s \in S with lim sup n ( P ( n ) ( s ) ) 1 / n = 1 \lim {\sup _n}{({P^{(n)}}(s))^{1/n}} = 1 . We study limit theorems for the convolution powers P ( n ) {P^{(n)}} of P P implied by the above property and further the class of all semigroups with this property. Theorem 3 relates this class of semigroups to left amenable semigroups.

Extreme Invariant Means without Minimal Support

Let $S$ be a left amenable semigroup. We show that if $S$ has a subset satisfying a certain condition, then there is an extreme left invariant mean on $S$ whose support is not a minimal closed invariant subset of $\beta S$. Then we show that all infinite solvable groups and countably infinite locally finite groups have such subsets.

Matrix summability in amenable semigroups

Necessary and sufficient conditions are given for an infinite matrix to be almost Schur and almost strongly regular in left amenable semigroups.

On Finite Invariant Measure for Semigroups of Operators

Let Σ be a left amenable semigroup, and let {Tσ: σ ∊ Σ} be a representation of Σ as a semigroup of positive linear contraction operators on L1(X, 𝓐, p). This paper is devoted to the study of existence of a finite equivalent invariant measure for such semigroups of operators.

On a Conjecture of E. Granirer Concerning the Range of an Invariant Mean

The purpose of this paper is to prove the following conjecture of E. Granirer: if $S$ is an infinite right cancellation left amenable semigroup then for each left invariant mean $\phi$ of $S$.

On a conjecture of E. Granirer concerning the range of an invariant mean

The purpose of this paper is to prove the following conjecture of E. Granirer: if S S is an infinite right cancellation left amenable semigroup then for each left invariant mean ϕ \phi of S S .

Notes on finite left amenable semigroups

Notes on finite left amenable semigroups

On the range of an invariant mean

Introduction. Let S be a discrete semigroup, m(S) the space of bounded real functions on S with the usual sup. norm, and m(S)* the conjugate Banach space of m(S). An element b e m(S)* is a mean if O(f) >0 whenever f>0 and #(1) =1, where 1 denotes also the constant one function on S. S is said to be left [right] amenable if there is a mean b E m(S)* which is in addition left [right] invariant, i.e., satisfies c(fa) = off) [5b(fa) = b(f)] for each f n m(S) and a e S (where fa(S) =f(as) and fa(S) =f(sa) for f in m(S) and a, s E S). S is amenable if there is a mean 0 E m(S)* which is left and right invariant. If Ac S, then 1A will be the function which is one on A and zero otherwise. We shall write 1 instead of ls and ?>(A) instead of k(lA) (if E m(S)*), sometimes. The range of an element f E m(S) is the set of numbers {+(A), where A ranges over all subsets of S}. It is clear that the range of a mean is a subset of [0, 1] ={x; O<x< l}. If S is a left amenable semigroup, define the following relation between elements of S: a b iff as = bs for some s in S. The relation is an equivalence relation which is two-sidedly stable (or a congruence), i.e., if a -.b, then asrbs and sasb for any s in S (since S is left amenable) (see [3, p. 371] and Ljapin [1, p. 39]). If s e S, let s' stand for the equivalence class to which s belongs and let S'={s'; s e S}. A multiplication between the elements of S' can be defined by s't'= (st)'. This multiplication is well defined and associative, rendering thus S' a semigroup (since is a congruence. See Ljapin [1, pp. 265-266]). Furthermore, S' has right cancellation (and coincides with S if S has right cancellation). It has been shown by this author in [4] (Corollary to Lemma 1) that, if S has right cancellation, is left amenable and contains an element of infinite order, then the range of any invariant mean on m(S) contains the set of rationals in [0, 1]. Moreover, if 0 < r < 1 is a rational number, then there is a set A (-S such that +(A) = r for any left invariant mean 0 on m(S). T. Mitchell has even suggested orally a proof to show that, for the above considered semigroup S, the range of each left invariant mean ? on m(S) is the whole [0, 1] interval. We would like to prove in this paper the following: CONJECTURE. Let S be a left amenable semigroup. Then the range of each left invariant mean on m(S) is the whole [0, 1] interval if and only if S' is infinite.