Left Invariant Mean Research Articles

Invariant means and multipliers on convolution quantum group algebras

Let [Formula: see text] be a locally compact quantum group. Then the space [Formula: see text] of trace class operators on [Formula: see text] is a Banach algebra with the convolution induced by the right fundamental unitary of [Formula: see text]. We show that properties of [Formula: see text] such as amenability, triviality and compactness are equivalent to the existence of left or right invariant means on the convolution Banach algebra [Formula: see text]. We also investigate the relation between the existence of certain (weakly) compact right and left multipliers of [Formula: see text] and some properties of [Formula: see text].

Skew-amenability of topological groups

We study skew-amenable topological groups, i.e., those admitting a left-invariant mean on the space of bounded real-valued functions left-uniformly continuous in the sense of Bourbaki. We prove characterizations of skew-amenability for topological groups of isometries and automorphisms, clarify the connection with extensive amenability of group actions, establish a Følner-type characterization, and discuss closure properties of the class of skew-amenable topological groups. Moreover, we isolate a dynamical sufficient condition for skew-amenability and provide several concrete variations of this criterion in the context of transformation groups. These results are then used to decide skew-amenability for a number of examples of topological groups built from or related to Thompson’s group F and Monod’s group of piecewise projective homeomorphisms of the real line.

Multi-cubic functional equations in Lipschitz spaces

In this paper, we prove stability of multi-cubic functional equations in Lipschitz spaces by property multi-symmetric left invariant mean. Indeed, we prove under certain Lipschitz condition a family of Lipschitz mappings can be approximated by multi-cubic mappings.

When is an invariant mean the limit of a Følner net?

Let G be a locally compact amenable group, TLIM(G) the topological left-invariant means on G, and TLIM0(G) the limit points of Følner-nets. I show that TLIM0(G)=TLIM(G) unless G is σ-compact non-unimodular, in which case TLIM0(G)≠TLIM(G). This improves a 1970 result of Chou and a 2009 result of Hindman and Strauss. I consider the analogous problem for the non-topological left-invariant means, and give a short construction of a net converging to invariance “weakly but not strongly,” simplifying the proof of a 2001 result of Rosenblatt and Willis. I conclude with a discussion of some open problems.

Convergence of almost orbits of semigroups

In this paper we establish the following general theorem. Let S be a semitopological reversible semigroup and let Y be a linear subspace of $${\ell }^\infty (S)$$ which contains the constants and is left translation invariant. Suppose that Y has a left invariant mean. Let $$(E, \Vert \cdot \Vert _E)$$ be a uniformly convex Banach space and let C be a nonempty, bounded, closed and convex subset of E. Assume that C has nonempty interior, is locally uniformly rotund and $$\mathcal {T}=\left\{ T\left( s\right) \right\} _{s \in S}$$ is an asymptotically nonexpansive semigroup which acts on C. Assume also that $$\mathcal {T}$$ has a unique fixed point $$\tilde{x}$$ and that, in addition, this point $$\tilde{x}$$ lies on the boundary $$\partial C$$ of C. Then $$\{u(s) \}_{s \in S}$$ converges strongly to $$\tilde{x}$$ for each almost orbit $$u \in AO_Y (\mathcal {T})$$ .

An amenability-like property of finite energy path and loop groups

We show that the groups of finite energy loops and paths (that is, those of Sobolev class H 1 ) with values in a compact connected Lie group, as well as their central extensions, satisfy an amenability-like property: they admit a left-invariant mean on the space of bounded functions uniformly continuous with regard to a left-invariant metric. Every strongly continuous unitary representation π of such a group (which we call skew-amenable) has a conjugation-invariant state on B(ℋ π ).

Invariant means on weakly almost periodic functions and generalized fixed point properties

In this paper, we prove common fixed point theorems for Generalized Suzuki Contractions (abbreviated as GSC) involving two semi-topological semigroups of self-mappings $$S_{1}$$ and $$S_{2}$$ , besides establishing the existence of a left invariant mean (abbreviated as LIM) on the space of all weakly almost periodic functions on $$S_{1}\cap S_{2}$$ (abbreviated as $$WAP(S_{1}\cap S_{2})$$ ).

Almost tri-cubic functions with Lipschitz condition

In this paper, we approximate a family of functions with a tri-symmetric left invariant mean property by tri-cubic functions in Lipschitz spaces.

On the amenability of a class of Banach algebras with application to measure algebra

Abstract Let 𝓛 be a Lau algebra and X be a topologically invariant subspace of 𝓛* containing UC(𝓛). We prove that if 𝓛 has a bounded approximate identity, then strict inner amenability of 𝓛 is equivalent to the existence of a strictly inner invariant mean on X. We also show that when 𝓛 is inner amenable the cardinality of the set of topologically left invariant means on 𝓛* is equal to the cardinality of the set of topologically left invariant means on RUC(𝓛). Applying this result, we prove that if 𝓛 is inner amenable and 〈𝓛2〉 = 𝓛, then the essential left amenability of 𝓛 is equivalent to the left amenability of 𝓛. Finally, for a locally compact group G, we consider the measure algebra M(G) to study strict inner amenability of M(G) and its relation with inner amenability of G.

Amenability and Hahn-Banach extension property for set-valued mappings

Amenability is an important notion in harmonic analysis on groups and semigroups, and their associated Banach algebras. In this paper, we present some characterization of a semitopological semigroup $S$ on the existence of a left invariant mean on $\text{\rm LUC}(S)$, $\text{\rm AP}(S)$ and $\text{\rm WAP}(S)$ in terms of Hahn-Banach extension theorem, which extend the first author's early results in 1970s. Moreover, we refine and extend the well known Day's result and Mitchell's results on fixed point properties for set-valued mappings. As an application, we give an application of our result to a class of Banach algebras related to amenability of groups and semigroups.

Introverted subspaces of the duals of measure algebras

Let $\mathcal{G} $ be a locally compact group. In continuation of our studies on the first and second duals of measure algebras by the use of the theory of generalized functions, here we study the C$^*$-subalgebra $GL_0(\mathcal{G})$ of $GL(\mathcal{G})$ as an introverted subspace of $M(\mathcal{G} )^*$. In the case where $\mathcal{G} $ is non-compact, we show that any topological left invariant mean on $GL(\mathcal{G} )$ lies in $GL_0(\mathcal{G} )^\perp $. We then endow $GL_0(\mathcal{G} )^*$ with an Arens-type product, which contains $M(\mathcal{G} )$ as a closed subalgebra and $M_a(\mathcal{G} )$ as a closed ideal, which is a solid set with respect to absolute continuity in $GL_0(\mathcal{G} )^*$. Among other things, we prove that $\mathcal{G} $ is compact if and only if $GL_0(\mathcal{G} )^*$ has a non-zero left (weakly) completely continuous element.

Common fixed point theorems for semigroup actions of Kannan’s type on strictly convex Banach spaces

Let K be a weakly compact subset of a strictly convex Banach space X. Let S be a semitopological semigroup which acts on K so that the action is weakly separately continuous of Kannan mappings with some additional conditions such that for every $$f \in C(K)$$ , the functions $$s \in S \longrightarrow f_x(s) = f(s.x)$$ belongs to Z a closed linear subspace of $$l^{\infty } (S)$$ containing constants and invariant under translations for every $$x \in K$$ . We prove that if Z has a left invariant mean then K has a common fixed point for S.

Stability of Multi-Quadratic Functions in Lipschitz Spaces

The algebra of Lipschitz functions on a complete metric space plays a role in non-commutative metric theory similar to that played by the algebra of continuous functions on a compact space in non-commutative topology. The notion of stability of functional equations was posed by Ulam, and then, Hyers gave the first significant partial solution in 1941. The subject has been established and developed by an increasing number of mathematicians in various spaces, particularly during the last two decades. In Lipschitz spaces, the notion of stability-type problems was introduced by Tabor and Czerwik. It should be mentioned that this subject area has been considered less attention over the recent years. In this paper, we prove stability of multi-quadratic functional equations in Lipschitz spaces by introducing the notion of a multi-symmetric left invariant mean. Indeed, we prove under certain Lipschitz conditions a family of Lipschitz functions can be approximated by multi-quadratic functions.

Amenability and Fan–Glicksberg theorem for set-valued mappings

In this paper, we begin by discussion of some well known results on the existence of left invariant means in the spaces: LUC(S), AP(S) and W AP(S) with Hahn-Banach extension theorem. We then give a new and precise proof of the well known Fan–Glicksberg fixed point theorem. This is then followed by a discussion on some related open problems.

Algebraic structure of semigroup compactifications: Pym's and Veech's Theorems and strongly prime points

The spectrum of an admissible subalgebra A(G) of LUC(G)(G), the algebra of right uniformly continuous functions on a locally compact group G, constitutes a semigroup compactification GA of G. In this paper we analyze the algebraic behaviour of those points of GA that lie in the closure of A(G)-sets, sets whose characteristic function can be approximated by functions in A(G). This analysis provides a common ground for far reaching generalizations of Veech's property (the action of G on GLUC(G) is free) and Pym's Local Structure Theorem. This approach is linked to the concept of translation-compact set, recently developed by the authors, and leads to characterizations of strongly prime points in GA, points that do not belong to the closure of G⁎G⁎, where G⁎=GA∖G. All these results will be applied to show that, in many of the most important algebras, left invariant means of A(G) (when such means are present) are supported in the closure of G⁎G⁎.

Invariantly complemented and amenability in Banach algebras related to locally compact groups

In this paper, among other things, we show that there is a close connection between the existence of a bounded projection on some Banach algebras associated to a locally compact group~$G$ and the existence of a left invariant mean on $L^\infty (G)$. A necessary and sufficient condition is found for a locally compact group to possess a left invariant mean.

Weakly almost periodic functionals on certain Banach algebras

For a Lau algebra A, we study the Banach space WAP(A) of all weakly almost periodic functionals on A to obtain some equivalent conditions for the existence of topological left invariant means on a topological left introverted subspace X of A contained in WAP(A). Finally, we consider relations between the existence of a topological left invariant mean on X and a common fixed point property.

Erratum to: Quartic functional equations in Lipschitz spaces

In [1] we proved the stability of quartic functional equations by using 2-variable quadratic functional equations inLipschitz spaces. In this paper,we show that the results presented in [1] only hold for quadratic functional equations and sowe correct our results and approximate the quartic functional equations by using biquadratic functional equations. Indeed, the equality K (2x, 2y) = 24K (x, y), used in [1, Theorem 2.1], does not hold for the 2-variable quadratic functional equation, that is, the function Q defined in [1, Theorem 2.1] is quadratic and it is not quartic. We need to introduce the notion of symmetric left invariant mean. Let G be an abelian group, E a vector space, and S(E) a family of subsets of E . By B(G, S(E))we denote the family of all functions f : G −→ E such that Im f ⊂ A for some A ∈ S(E). We say that B(G, S(E)) admits a symmetric left invariant mean (briefly SLIM), if the family S(E) is linearly invariant and there exists a linear operatorM : B(G, S(E)) −→ E such that (i) if fx,y ∈ B(G, S(E)) and (x, y) ∈ G × G, then M[ fx,y] = M[ fy,x ],

A

We give a complete characterization of connected Lie groups with the Approximation Property for groups (AP). To this end, we introduce a strengthening of property (T), that we call property (T*), which is a natural obstruction to the AP. In order to define property (T*), we first prove that for every locally compact group G, there exists a unique left invariant mean on the space of completely bounded Fourier multipliers of G. A locally compact group G is said to have property (T*) if this mean is a weak* continuous functional. After proving that the groups SL(3,R), Sp(2,R), and the universal covering of Sp(2,R) have property (T*), we address the question which connected Lie groups have the AP. A technical problem that arises when considering this question from the point of view of the AP is that the semisimple part of the global Levi decomposition of a connected Lie group need not be closed. Because of an important permanence property of property (T*), this problem vanishes. It follows that a connected Lie group has the AP if and only if all simple factors in the semisimple part of its Levi decomposition have real rank 0 or 1. Finally, we are able to establish property (T*) for all connected simple higher rank Lie groups with finite center.

Fixed point properties for semigroups of nonlinear mappings on unbounded sets

A well-known result of W. Ray asserts that if C is an unbounded convex subset of a Hilbert space, then there is a nonexpansive mapping T: C→C that has no fixed point. In this paper we establish some common fixed point properties for a semitopological semigroup S of nonexpansive mappings acting on a closed convex subset C of a Hilbert space, assuming that there is a point c∈C with a bounded orbit and assuming that certain subspace of Cb(S) has a left invariant mean. Left invariant mean (or amenability) is an important notion in harmonic analysis of semigroups and groups introduced by von Neumann in 1929 [28] and formalized by Day in 1957 [5]. In our investigation we use the notion of common attractive points introduced recently by S. Atsushiba and W. Takahashi.