Left Invariant Mean Research Articles

Equivalent Types of Invariant Means on Locally Compact Groups

For $G$ a locally compact amenable group, we establish the equivalence of left invariant means and topologically left invariant means on ${L^\infty }(G)$.

Equivalent types of invariant means on locally compact groups

For G G a locally compact amenable group, we establish the equivalence of left invariant means and topologically left invariant means on L ∞ ( G ) {L^\infty }(G) .

Invariant means on locally compact semigroups

Let G be a locally compact semigroup (jointly continuous semigroup operation), M(G) the algebra of all bounded regular Borel measures on G (with convolution as multiplication), E a separated locally convex space and S a compact convex subset of E. We show that there is a left invariant mean on the space LUC(G) of all bounded left uniformly continuous functions on G iff G has the following fixed point property: For any bilinear mapping T: M(G) x E E (denoted by (,a, s) -* T,,(s)) such that (a) T,,(S) C S for any It > O, Il,ull = 1, (b) TMu*v = Tu ? Tv for any It, vCM(G), (c) T,u:S-S is continuous for any ,u _ O, I,ul = 1, and (d) It TMu(s) is continuous for each s C S when M(G) has the topology induced by the seminorms pf(,U) = Sf f dyl, f C LUC(G), there is some so C S such that TM(so) = so for any I > 0, Ilufl = 1.

On a Geometric Property of the Set of Invariant Means on a Group

If G is a discrete group and $x \in G$ then $x^\sim$ denotes the homeomorphism of $\beta G$ onto $\beta G$ induced by left multiplication by x. A subset K of $\beta G$ is said to be invariant if it is closed, nonempty and $x^\sim \emptyset K \subset K$ for each $x \in G$. Let $ML(G)$ denote the set of left invariant means on G. (They can be considered as measures on $\beta G$.) Let G be a countably infinite amenable group and let K be an invariant subset of $\beta G$. Then the nonempty ${w^ \ast }$-compact convex set $M(G,K) = \{ \phi \in ML(G):{\text {suppt}}\phi \subset K\}$ has no exposed points (with respect to ${w^ \ast }$-topology). Therefore, it is infinite dimensional.

On a geometric property of the set of invariant means on a group

If G is a discrete group and x ∈ G x \in G then x ∼ x^\sim denotes the homeomorphism of β G \beta G onto β G \beta G induced by left multiplication by x. A subset K of β G \beta G is said to be invariant if it is closed, nonempty and x ∼ ∅ K ⊂ K x^\sim \emptyset K \subset K for each x ∈ G x \in G . Let M L ( G ) ML(G) denote the set of left invariant means on G. (They can be considered as measures on β G \beta G .) Let G be a countably infinite amenable group and let K be an invariant subset of β G \beta G . Then the nonempty w ∗ {w^ \ast } -compact convex set M ( G , K ) = { ϕ ∈ M L ( G ) : suppt ϕ ⊂ K } M(G,K) = \{ \phi \in ML(G):{\text {suppt}}\phi \subset K\} has no exposed points (with respect to w ∗ {w^ \ast } -topology). Therefore, it is infinite dimensional.

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 the Size of the Set of Left Invariant Means on a Semigroup

On the Size of the Set of Left Invariant Means on a Semigroup

Amenable actions of locally compact groups

We consider a locally compact group G with jointly continuous action G × Z → Z on a locally compact space. The finite Radon (regular Borel) measures M( G) act naturally on various function spaces supported on Z, including the continuous bounded functions CB( Z). If Z supports a (not necessarily unique) nonnegative quasi-invariant measure ν we apply some recent studies [ 7] to define a Banach module action of M( G) on L 1( Z, ν), and this induces a natural adjoint action on L ∞(Z, ν) = L 1(Z, ν) ∗ . These actions of M( G) (and of G) give us definitions of amenability of the action of G on various function spaces supported on Z, including CB( Z) and L ∞( Z, ν) (when there is a quasi-invariant ν on Z), corresponding to the existence of a left-invariant mean on these various spaces. When G acts on one of its coset spaces solG H , H a closed subgroup, there is always a quasi-invariant measure on G H and the different definitions of amenability of the action G × G H → G H all coincide. A number of manipulations of invariant means on groups (the case where G = Z) then carry over to the context of transformation groups. We apply them to explore the following problems. Let G × G H → G H be an amenable action of G on one of its coset spaces. Let ν be a quasi-invariant measure on G H .

Functional analytic properties of extremely amenable semigroups

Introduction. Let S be a semigroup m(S) the Banach space of all bounded real functions on S with the norm lf l =sup {ff(s)I; s E S} and m(S)* the conjugate Banach space of m(S). 1 E m(S)* is a mean if > O (i.e. +(f)O> iff O, fe m(S)) and q(1) = 1 (1 stands also for the constant one function on S). The semigroup S is left amenable (LA) if there exists a mean 1 E m(S)* which is left invariant i.e. q(fa) = +(f) for all a E S, f r m(S) (we define fa(S) =f(as), fa(s) =f(sa), laf=fa, raf=fa, La = la*: m(S)* m> m(S)*, Ra = r* for all a, s E S and f cr m(S)). The semigroup S is extremely left amenable (ELA) if there is a left invariant mean 1 on m(S) which is multiplicative (i.e. such that q(fg) = q(f)b(g) for all f, g Er m(S)). In anology one defines (extremely) right amenable semigroups (ERA) RA. ELA semigroups have been studied at first by T. Mitchell in [22] (under different terminology) and afterwards by this author in [15], [16]. They are known to be characterized by the following equivalent properties: 1. S is ELA. 2. S has the common fixed point property on compacta (i.e. for any compact Hausdorff space and any homomorphic representation S' of S as a semigroup of continuous maps from X to X (under functional composition), there is some x0 E X such that s'(x0)=x0 for all s' E S' (Mitchell [22]). 3. For any a, b E S there is some c E S with ac= bc = c [16]. 4. For any function h of the ideal

On the size of the set of left invariant means on a semi-group

On the size of the set of left invariant means on a semi-group

Ratio properties in locally compact amenable groups

0. Introduction and summary. Let r be a locally compact (Hausdorff) topological group, and let 1*1 denote a fixed left Haar measure on the Borel sets of 1' (in the sense of [1, ?51]). It is known (see [2]) that the existence of a left invariant mean on LP(r) is equivalent to the Folner Condition (FC): To any E > 0 and any compact set Cc r, there corresponds a Borel set V of finite nonzero measure such that

Covering properties and F�lner conditions for locally compact groups

LetG be a locally compact group with left Haar measurem G on the Borel sets IB(G) (generated by open subsets) and write |E|=m G (E). Consider the following geometric conditions on the groupG. (FC If ɛ>0 and compact setK⊂G are given, there is a compact setU with 0 0 and compact setK⊂G, which includes the unit, are given there is a compact setU with 0<|U|<∞ and |K U ΔU|/|U|<ɛ. HereA ΔB=(A/B)⌣(B/A) is the symmetric difference set; by regularity ofm G it makes no difference if we allowU to be a Borel set. It is well known that (A)⇒(FC) and it is known that validity of these conditions is intimately connected with “amenability” ofG: the existence of a left invariant mean on the spaceCB(G) of all continuous bounded functions. We show, for arbitrary locally compact groupsG, that (amenable)⇔(FC)⇔(A). The proof uses a covering property which may be of interest by itself: we show that every locally compact groupG satisfies. (C) For at least one setK, with int(K)≠O and $$\bar K$$ compact, there is an indexed family {x α∶αeJ}⊂G such that {Kx α} is a covering forG whose covering index at each pointg (the number of αeJ withgeKx α) is uniformly bounded throughoutG.

A Mixing Condition for Extreme Left Invariant Means

A Mixing Condition for Extreme Left Invariant Means

Fixed Points and Multiplicative Left Invariant Means

Fixed Points and Multiplicative Left Invariant Means

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.

Fixed points and multiplicative left invariant means

Fixed points and multiplicative left invariant means

A mixing condition for extreme left invariant means

A mixing condition for extreme left invariant means

Constant Functions and Left Invariant Means on Semigroups

Constant Functions and Left Invariant Means on Semigroups

Constant functions and left invariant means on semigroups

semigroup, is left amenable, then K contains a common fixed point of the fanmily S. Proof. Since S is left amenable, then by Theorem 3, there exists a net, {T1} where T. E P, such that for any fe En(S), {T,,f} converges pointwise to a constant function. For any T,, there exists 0, E 'D such that T, = E, s4,(s)r.. Let the map J,: KK be given by J,,k = Y, E s4,(s)s(k), for k E K. For the remainder of the proof, let y be a specific point in K. By compactness of K, there exists a subnet, {J,} of {J,}, such that {J,6y} converges to some yo E K. And the associated subnet, {T,} of {T}, satisfies that for anyfe rm(S), {Tjf} converges pointwise to a constant function, since {TJf} is a subnet of {Tj}. We will show that for any so E S, that soyo is the required common fixed point. (It can be shown by counterexample that yo itself need not be one.) For each tE X*, define as in [2, p. 587], a real valued function, f,E on S byf,E(s) = ,(sy), for s E S. Since ,u is continuous over the compact set K, then f,E e m(S). So for s' eS, (TJ ,1)s' = OAS) ((rsf) ) = O ?As)f(s 's) = E A(s)p(s 'sy) s e S s e S (s ( S SY) =l(S (sS(S)s)) = 1(s'J6y), by linearity of ,u and affineness of s', in steps 4 and 5, respectively. Then lim (T,6f,)s' = lim (r(s'(J6y))) = ,t(s' ( lim (J,6Y)))

A theorem on amenable semigroups

where (laf)(g)=f(ag) and (raf)(g)=f(ga). An invariant mean is a right and left invariant mean. Ml(G), [Mr(G)] c m(G)* will denote the set of left [right] invariant means and dim MIG = n will mean that the linear manifold spanned by Ml(G) c m(G)* is n-dimensional (see [5, ?2]). Q: 1l(G) -+ m(G)* will denote the natural mapping of the semigroup algebra 11(G) into m(G)*. The following is a result of I. S. Luthar (see [9]): A commutative semigroup G has a unique invariant mean (i.e., dim Ml(G) = 1) if and only if G contains a finite ideal. Luthar's proof also yields that the unique invariant mean has to belong to Q11(G). The main result of this paper is the following: