Left Invariant Mean Research Articles

A Fixed Point Theorem and the Existence of a Haar Measure for Hypergroups Satisfying Conditions Related to Amenability

AbstractIn this paperwe present a fixed point property for amenable hypergroups that is analogous to Rickert’s fixed point theorem for semigroups. It equates the existence of a left invariant mean on the space of weakly right uniformly continuous functions to the existence of a fixed point for any action of the hypergroup. Using this fixed point property, certain hypergroups are shown to have a left Haar measure.

Vanishing of ℓp-cohomology and transportation cost

In this paper, it is shown that the reduced ℓ p -cohomology is trivial for a class of finitely generated amenable groups called transport amenable. These groups are those for which there exists a sequence of measures ξ n converging to a left-invariant mean and such that the transport cost between ξ n displaced by multiplication on the right by a fixed element and ξ n is bounded (uniformly in n). This class contains groups with controlled Fø lner sequences (such as polycyclic groups) as well as some wreath products (such as H ' ≀ H where H is finitely generated Abelian and the ‘lamp state’ group H ' is finitely generated amenable).

Invariant Means and Reversible Semigroup of Relatively Nonexpansive Mappings in Banach Spaces

The purpose of this paper is to study modified Halpern type and Ishikawa type iteration for a semigroup of relatively nonexpansive mappingsI={T(s):s∈S}on a nonempty closed convex subsetCof a Banach space with respect to a sequence of asymptotically left invariant means{μn}defined on an appropriate invariant subspace ofl∞(S), whereSis a semigroup. We prove that, given some mild conditions, we can generate iterative sequences which converge strongly to a common element of the set of fixed pointsF(I), whereF(I)=⋂{F(T(s)):s∈S}.

Approximating fixed points for a reversible semigroup of Lipschitzian mappings in a smooth Banach space

In this paper, we approximate a fixed point of the semigroup of Lipschitzian mappings from a nonempty compact convex subset C of a smooth Banach space E into C with a uniform Lipschitzian condition and with respect to a finite family of sequences of left strong regular invariant means defined on an appropriate invariant subspace of . Our result extends the main results announced by several others. MSC:47H09, 47H10, 47J25.

Syndetic sets and amenability

We prove that if an infinite, discrete semigroup has the property that every right syndetic set is left syndetic, then the semigroup has a left invariant mean. We prove that the weak ∗ * -closed convex hull of the two-sided translates of every bounded function on an infinite discrete semigroup contains a constant function. Our proofs use the algebraic properties of the Stone-Cech compactification.

Convergence of a hybrid algorithm for a reversible semigroup of nonlinear operators in Banach spaces

The purpose of this paper is to study hybrid iterative schemes of Halpern types for a semigroup ℑ = { T ( s ) : s ∈ S } of relatively nonexpansive mappings on a closed and convex subset C of a Banach space with respect to a sequence { μ n } of asymptotically left invariant means defined on an appropriate invariant subspace of l ∞ ( S ) . We prove that given a certain sequence { α n } in [ 0 , 1 ] , x ∈ C , we can generate an iterative sequence { x n } which converges strongly to Π F ( ℑ ) x where Π F ( ℑ ) x is the generalized projection from C onto the fixed point set F ( ℑ ) . Our main result is even new for the case of a Hilbert space.

Uniform continuity over locally compact quantum groups

We define, for a locally compact quantum group 𝔾 in the sense of Kustermans–Vaes, the space of LUC(𝔾) of left uniformly continuous elements in L∞(𝔾). This definition covers both the usual left uniformly continuous functions on a locally compact group and Granirer's uniformly continuous functionals on the Fourier algebra. We show that LUC(𝔾) is an operator system containing the C*-algebra 𝒷0(𝔾) and contained in its multiplier algebra ℳ(𝒷0(𝔾)). We use this to partially answer an open problem by Bédos–Tuset: if 𝔾 is co-amenable, then the existence of a left invariant mean on ℳ(𝒷0(𝔾)) is sufficient for 𝔾 to be amenable. Furthermore, we study the space WAP(𝔾) of weakly almost periodic elements of L∞(𝔾): it is a closed operator system in L∞(𝔾) containing 𝒷0(𝔾) and, for co-amenable 𝔾, contained in LUC(𝔾). Finally, we show that, under certain conditions, which are always satisfied if 𝔾 is a group, the operator system LUC(𝔾) is a C*-algebra.

Density and invariant means in left amenable semigroups

A left cancellative and left amenable semigroup S satisfies the Strong Følner Condition. That is, given any finite subset H of S and any ϵ > 0 , there is a finite nonempty subset F of S such that for each s ∈ H , | s F △ F | < ϵ ⋅ | F | . This condition is useful in defining a very well behaved notion of density, which we call Følner density, via the notion of a left Følner net, that is a net 〈 F α 〉 α ∈ D of finite nonempty subsets of S such that for each s ∈ S , ( | s F α △ F α | ) / | F α | converges to 0. Motivated by a desire to show that this density behaves as it should on cartesian products, we were led to consider the set LIM 0 ( S ) which is the set of left invariant means which are weak ∗ limits in l ∞ ( S ) ∗ of left Følner nets. We show that the set of all left invariant means is the weak ∗ closure of the convex hull of LIM 0 ( S ) . (If S is a left amenable group, this is a relatively old result of C. Chou.) We obtain our desired density result as a corollary. We also show that the set of left invariant means on ( N , + ) is actually equal to LIM 0 ( N ) . We also derive some properties of the extreme points of the set of left invariant means on S, regarded as measures on βS, and investigate the algebraic implications of the assumption that there is a left invariant mean on S which is non-zero on some singleton subset of βS.

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.

INVARIANT MEANS AND FIXED POINT PROPERTIES OF SEMIGROUP OF NONEXPANSIVE MAPPINGS

This paper outlines some of my recent joint works with Q. Takahashi on fixed point properties or ergodic properties for semigroup of nonexpansive mappings on closed convex subsets of a Banach space and their relationship with existence of left invariant mean on certain subspaces of bounded realvalued functions on the semigroup.

Strongly and weakly almost periodic linear maps on semigroup algebras

Let S be a foundation locally compact topological semigroup. Two new topologies τ c and τ w are introduced on M a (S)*. We introduce τ c and τ w almost periodic functionals in M a (S)*. We study these classes and compare them with each other and with the norm almost periodic and weakly almost periodic functionals. For f∈M a (S)*, it is proved that T f ∈ℬ(M a (S),M a (S)*) is strong almost periodic if and only if f is τ c -almost periodic. Indeed, we have obtained a generalization of a well known result of Crombez for locally compact group to a more general setting of foundation topological semigroups. Finally if P(S) (the set of all probability measures in M a (S)) has the semiright invariant isometry property, it is shown that the set of τ w -almost periodic functionals has a topological left invariant mean.

Fixed points of non-expansive mappings associated with invariant means in a Banach space

In this paper, we study the fixed point set of the non-expansive mapping T μ for a Banach space with uniformly Gâteaux differentiable norm when μ is a multiplicative left invariant mean on l ∞ ( S ) . As an application, we establish nonlinear ergodic properties for an extremely amenable semigroup of non-expansive mappings in a Banach space with uniformly Gâteaux differentiable norm. Furthermore, we improve a recent result of Atsushiba and Takahashi [S. Atsushiba, W. Takahashi, Weak and strong convergence theorems for non-expansive semigroups in a Banach spaces satisfying Opial’s condition, Sci. Math. Jpn. (in press)] on the fixed point set of non-expansive mappings associated with a left invariant mean on a left amenable semigroup.

Topologically left invariant means on semigroup algebras

LetM(S) be the Banach algebra of all bounded regular Borel measures on a locally compact Hausdorff semitopological semigroupS with variation norm and convolution as multiplication. We obtain necessary and sufficient conditions forM(S)* to have a topologically left invariant mean.

Nonlinear submeans on semigroups

The purpose of this paper is to study some algebraic structure of submeans on certain spaces $X$ of bounded real valued functions on a semigroup and to find local conditions on $X$ in terms of submean for the existence of a left invariant mean.

On the Space of Weakly Almost Periodic Functionals and Its Amenability

Let S be a locally compact semitopological semigroup with measure algebra M(S), M0(S) the set of all probability measures in M(S) and WF(S) the space of weakly almost periodic functionals on M(S)*. Assuming that M0(S) has the semiright invariant isometry property, it is shown that WF(S) has a topological left invariant mean (TLIM) whenever the center of M0(S) is nonempty; in particular if either the center of S is nonempty or S has a left identity, then WF(S) has a TLIM. Finally if, for each μ ∈ M0(S), the mapping v → v * μ of M0(S) into itself is surjective and the center of M0(S) is nonempty, then WF(S) has a TLIM. We also generalize some results from discrete case to topological one.

Cohomology on hypergroup algebras

There are concepts which are related to or can be formulated by homological techniques, such as derivations, multipliers and lifting problems. Moreover, a Banach algebra A is said to be amenable if H1(A,X*)=0 for every A-dual module X*. Another concept related to the theory is the concept of amenability in the sense of Johnson. A topological group G is said to be amenable if there is an invariant mean on L 8(G). Johnson has shown that a topological group is amenable if and only if the group algebra L1(G) is amenable. The aim of this research is to define the cohomology on a hypergroup algebra L(K) and extend the results of L1(G) over to L(K). At first stage it is viewed that Johnson's theorem is not valid so more. If A is a Banach algebra and h is a multiplicative linear functional on A, then (A,h) is called left amenable if for any Banach two-sided A-module X with ax=h(a)x(a? A, x? X),H1(A,X*)=0. We prove that (L(K),h) is left amenable if and only if K is left amenable. Where, the latter means that there is a left invariant mean m on C(K), i. e., m(lf)=m(f)x, where lxf(µ)=f(dx*µ). In this case we briefly say that L(K) is left amenable. Johnson also showed that L1(G) is amenable if and only if the augmentation ideal I={f? L1(G)|∫Gf=0}0 has abounded right approximate identity. We extend this result to hypergroups.

The exposed points of the set of invariant means on an ideal

Let G G be a σ \sigma -compact locally compact nondiscrete group and let Q Q be a G G -invariant ideal of L ∞ ( G ) L^{\infty }(G) . We denote the set of left invariant means m m on L ∞ ( G ) L^{\infty }(G) that are zero on Q Q (i.e. m ( f ) = 0 m(f) = 0 for all f ∈ Q f\in Q ) by L I M Q LIM_{Q} . We show that, when G G is amenable as a discrete group and the closed G G -invariant subset of the spectrum of L ∞ ( G ) L^{\infty }(G) corresponding to Q Q is a G δ G_{\delta } -set, L I M Q LIM_{Q} is very large in the sense that every nonempty G δ G_{\delta } -subset of L I M Q LIM_{Q} contains a norm discrete copy of β N \beta \mathbb {N} , where β N \beta \mathbb {N} is the Stone- C ˇ e c h \mathrm {\check {C}ech} compactification of the set N \mathbb {N} of positive integers with the discrete topology. In particular, we prove that L I M Q LIM_{Q} has no exposed points in this case and every nonempty G δ G_{\delta } -subset of the set of left invariant means on L ∞ ( G ) L^{\infty }(G) contains a norm discrete copy of β N \beta \mathbb {N} .

Locally Compact Groups, Invariant Means and the Centres of Compactifications

This paper brings together two apparently unrelated results about locally compact groups G by giving them a common proof. The first concerns the number of topologically left invariant means on L ∞ ( G ), while the second states that the topological centre of the largest semigroup compactification of G is simply G itself. On the way, we introduce as vital tools some new compactifications of the half line ([0, ∞), +), we produce a right invariant pseudometric on a compactly generated G for which the bounded sets are precisely the relatively compact sets, and we receive striking confirmation that some algebraic properties of semigroups can be transferred by maps which are quite far from being homomorphisms.

Amenability of Hopf von Neumann Algebras and Kac Algebras

Let (M,Γ) be a Hopf von Neumann algebra. The operator predualM*ofMis a completely contractive Banach algebra with multiplicationm=Γ*:M*⊗M*→M*. We call (M,Γ)operator amenableif the completely contractive Banach algebraM*isoperator amenable, i.e., for every operatorM*-bimoduleV, every completely bounded derivation fromM*into the dualM*-bimoduleV* is inner. There is a weaker notion of amenability introduced by D. Voiculescu. We say that a Hopf von Neumann algebra (M,Γ) isVoiculescu amenableif there exists a left invariant mean onM. We show that if a Hopf von Neumann algebra (M,Γ) is operator amenable, then it is Voiculescu amenable.For Kac algebras, there is astrong Voiculescu amenability. We show that for discrete Kac algebras, these amenabilities are all equivalent. In fact, if we letK=(M,Γ,κ,ϕ) be a discrete Kac algebra and letK=(M,Γ,κ,ϕ) be its (compact) dual Kac algebra, then the following are equivalent: (1)Kis operator amenable; (2)Kis Voiculescu amenable; (3) The von Neumann algebraMis hyperfinite; (4)Kis strong Voiculescu amenable; (5)Kis operator amenable; (6)M*has a bounded approximate identity.

A fixed point theorem for nonexpansive semigroups in p-uniformly convex Banach spaces

We prove that if RUC(S) has a left invariant mean,ρ = {Ts:s ∈S} is a continuous representation ofS as nonexpansive mappings on a closed convex subsetC of a p-uniformly convex and p-uniformly smooth Banach space andC contains an element of bounded orbit, thenC contains a common fixed point forρ.