A Relation Between Invariant Means on Lie Groups and Invariant Means on Their Discrete Subgroups

Introduction.M. M. Day has shown that an abelian group has only one invariant mean if, and only if, the group is finite [3, Theorem 4, p. 534].Knowing this we now ask to what extent the invariant means on a given abelian group are alike.In this paper we shall show that they all have the same "measure-theoretic character."An invariant mean on an abelian group G is a normalized, positive, translation-invariant linear functional defined on the space of all bounded complex-valued functions on G.There is a canonical way of associating with the invariant mean L a (unique) positive measure (which is invariant in a sense that will be made clear) defined on the Stone-Cech compactification of G (discrete topology).Using this one-to-one correspondence, we shall ascribe to the invariant mean L the measure-theoretic properties of .Thus the measure algebra of the invariant mean L is simply the measure algebra of the associated measure .Our main theorem (Theorem 2) is that the measure algebras of any two invariant means on G are isomorphic.A corollary extends this result to the measure algebras of invariant means on a subsemigroup of an abelian group.An essential lemma in the proof of the main theorem is derived in 2.We show that if m is the cardinality of G and is assumed to be infinite, then 2m is the dimension of the S2 space associated with an invariant mean on G.Because we believe this result to be of independent interest, we state it as Theorem 1.A corollary to Theorem 1 gives a similar result for invariant means on certain abelian semigroups.In particular, this corollary is applicable to the semigroup of positive integers.An invariant mean on this semigroup is called a Banach mean.The corollary shows that the S2 space associated with a Banach mean is not separable.The question of the separability of this S2 space was raised by P. Porcelli.As a final topic we shall discuss (in 4) some related problems.This discussion amounts to an exploration of further possibilities in the directions of Theorems 1 and 2, and we shall raise several questions in this connection.

Necessary and sufficient conditions are given for an infinite matrix to be almost Schur and almost strongly regular in left amenable semigroups. 1. Introduction.In [7] the author discusses various methods of matrix summability in amenable semigroups.In that paper, sufficient conditions were given for an infinite matrix to be almost Schur and almost strongly regular.Examples were given to show that the conditions are not necessary.Recently, P. Schaefer gave necessary and sufficient conditions for an infinite matrix to be almost strongly regular for the additive semigroup of positive integers [8].This was also obtained by Howard T. Bell via a different approach [1].In this paper, we give necessary and sufficient conditions for an infinite matrix to be almost Schur and almost strongly regular in certain amenable semigroups. Preliminaries.We shall freely use notations and definitions in [7].Recall that a semigroup S is left amenable (LA) if the Banach space of all bounded real-valued functions on 5 with the sup norm, m(S), has a normalized positive left translation invariant linear functional.Such a linear functional is called a left invariant mean (LIM).The semigroup is said to be extremely left amenable (ELA) if the LIM is also multiplicative.It is shown by Day [2, p. 524, Theorem 1] that if S is LA, there is a net {<f>x} of finite means converging to left invariance in norm.If, in addition, S is countable then this net can be replaced by a sequence [6, p. 42, Lemma 5.1].Examples of LA semigroups are commutative semigroups, solvable groups and locally finite groups.Extremely left amenable semigroups are precisely those in which every two elements have a common right zero.For more details, we refer the reader to [2], [3], [4], [5].

We study growth of 1-cocycles of locally compact groups, with values in unitary representations. Discussing the existence of 1-cocycles with linear growth, we obtain the following alternative for a class of amenable groups G containing polycyclic groups and connected amenable Lie groups: either G has no quasi-isometric embedding into Hilbert space, or G admits a proper cocompact action on some Euclidean space. On the other hand, noting that almost coboundaries (i.e. 1-cocycles approximable by bounded 1-cocycles) have sublinear growth, we discuss the converse, which turns out to hold for amenable groups with "controlled" Folner sequences; for general amenable groups we prove the weaker result that 1-cocycles with sufficiently small growth are almost coboundaries. Besides, we show that there exist, on a-T-menable groups, proper cocycles with arbitrary small growth.

Let $S$ be a topological semigroup such that the almost periodic functions on $S$ have a left invariant mean (this is the case, for example, when $S$ has finite intersection property for closed right ideals). Then whenever $S$ acts equicontinuously on a compact Hausdorff space $X$, there exists a compact group $G$ of homeomorphisms acting equicontinuously on a retract $Y$ of $X$ such that $S$ has a common fixed point in $X$ if and only if $G$ has a common fixed point in $Y$. This result generalises some recent work of T. Mitchell. As an application, we show that whenever $S$ acts equicontinuously on the closed unit interval $I$, then $I$ contains a common fixed point for $S$.

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.

Let S S be a topological semigroup such that the almost periodic functions on S S have a left invariant mean (this is the case, for example, when S S has finite intersection property for closed right ideals). Then whenever S S acts equicontinuously on a compact Hausdorff space X X , there exists a compact group G G of homeomorphisms acting equicontinuously on a retract Y Y of X X such that S S has a common fixed point in X X if and only if G G has a common fixed point in Y Y . This result generalises some recent work of T. Mitchell. As an application, we show that whenever S S acts equicontinuously on the closed unit interval I I , then I I contains a common fixed point for S S .

Let 5 be a discrete semigroup, P a probability measure on S and se S with lim sup"(P'",(j))1'" = l.We study limit theorems for the convolution powers P,n' ofP 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. Introduction.Let S be a discrete semigroup and P a probability measure on S, that is a real valued function on S with P(s)~^.0 for all seS and 2sesF(i)=l.Kesten ([4], [5]) characterized amenable groups by means of the asymptotic behavior of convolution powers of symmetric probability measures defined on the group.A more precise information for the asymptotic behavior was obtained in [2] and [3] for symmetric probability measures on a discrete amenable group.In what follows we will derive similar theorems for probability measures on discrete semigroups.Let S be a discrete semigroup, P a probability measure on S. Then Supp P={s/P(s)>Q} denotes the support of P. To say Supp P generates S means: S= \J%=1 (SuppP)n.For probability measures P, Q on S define their convolution P * Q by P * Q(s) = 2 WOW SlS2=S (the summation is to be extended over all representations ois as a product of two elements sx, s2 of S).P * Q is again a probability measure and Supp P * g = (Supp P) (Supp Q).We often write P=P,P(") = P * /><-i>.Kesten obtained the following characterization of discrete amenable groups:Let G be a discrete group with unit element e, P a symmetric probability measure on G (P(g)=P(g~1) for all g in G) such that G is generated by

We define a category containing the discrete quantum groups (and hence the discrete groups and the duals of compact groups) and the compact quantum groups (and hence the compact groups and the duals of discrete groups). The dual of an object can be defined within the same category and we have a biduality theorem. This theory extends the duality between compact quantum groups and discrete quantum groups (and hence the one between compact abelian groups and discrete abelian groups). The objects in our category are multiplier Hopf algebras, with invertible antipode, admitting invariant functionals (integrals), satisfying some extra condition (to take care of the non-abelianness of the underlying algebras). If we start with a multiplier Hopf ∗-algebra with positive invariant functionals, then also the dual is a multiplier Hopf ∗-algebra with positive invariant functionals. This makes it possible to formulate this duality also within the framework of C∗-algebras.

Let $K$ be a locally compact hypergroup with left Haar measure and let $L^1(K)$ be the complex Lebesgue space associated with it. Let $L^infty(K)$ be the dual of $L^1(K)$. The purpose of this paper is to present some necessary and sufficient conditions for $L^infty(K)^*$ to have a topologically left invariant mean. Some characterizations of amenable hypergroups are given.

We prove that the first reduced cohomology with values in a mixing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup></mml:math>-representation, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&lt;</mml:mo><mml:mi>p</mml:mi><mml:mo>&lt;</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math>, vanishes for a class of amenable groups including connected amenable Lie groups. In particular this solves for this class of amenable groups a conjecture of Gromov saying that every finitely generated amenable group has no first reduced <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>ℓ</mml:mi> <mml:mi>p</mml:mi> </mml:msup></mml:math>-cohomology. As a byproduct, we prove a conjecture by Pansu. Namely, the first reduced <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup></mml:math>-cohomology on homogeneous, closed at infinity, Riemannian manifolds vanishes. We also prove that a Gromov hyperbolic geodesic metric measure space with bounded geometry admitting a bi-Lipschitz embedded 3-regular tree has non-trivial first reduced <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup></mml:math>-cohomology for large enough <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math>. Combining our results with those of Pansu, we characterize Gromov hyperbolic homogeneous manifolds: these are the ones having non-zero first reduced <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup></mml:math>-cohomology for some <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>1</mml:mn><mml:mo>&lt;</mml:mo><mml:mi>p</mml:mi><mml:mo>&lt;</mml:mo><mml:mi>∞</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:math>

We prove that for actions of a discrete countable residuallyfinite amenable group on a compact metric space with specification property, periodic measures are dense in theset of invariant measures. We also prove that certain expansiveactions of a countable discrete group by automorphisms of compact abelian groups have specification property.

We investigate ten-dimensional gauge theories whose extra six-dimensional space is a compact coset space, $S/R$, and whose gauge group is a direct product of two Lie groups. We list candidates of the gauge group and embeddings of $R$ into them. After dimensional reduction of the coset space, we find fermion and scalar representations of ${G}_{\mathrm{GUT}}\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ with ${G}_{\mathrm{GUT}}=\mathrm{SU}(5)$, $\mathrm{SO}(10)$ and ${E}_{6}$, which accommodate all of the standard model particles. We also discuss possibilities to generate distinct Yukawa couplings among the generations using representations with different dimensions for ${G}_{\mathrm{GUT}}=\mathrm{SO}(10)$ and ${E}_{6}$ models.

This chapter is devoted to studying some concepts that will be extensively used in the last chapters, namely the cohomology of Lie algebras with values in a vector space, the Whitehead lemmas and Lie algebra extensions (which are related to second cohomology groups). The same three different cases of extensions of chapter 5 as well as the ℱ( M )-valued version of cohomology will be considered. In fact, the relation between Lie group and Lie algebra cohomology will be explored here, first with the simple example of central extensions of groups and algebras (governed by twococycles), and then in the higher order case, providing explicit formulae for obtaining Lie algebra cocycles from Lie group ones and vice versa. The Lie algebra cohomology a la Chevalley-Eilenberg, which uses invariant forms on a Lie group, is also presented in this chapter. This will turn out to be specially useful in the construction of physical actions (chapter 8), i.e . in the process of relating cohomology and mechanics. The BRST formulation of Lie algebra cohomology will also be discussed here due to its importance in gauge theories. Cohomology of Lie algebras: general definitions We now discuss the cohomology of Lie algebras with values on a vector space V following the same pattern as used in chapter 5 for the cohomology of groups with values in an abelian group. Let G be a Lie algebra, and let the vector space V over the field K be a ρ(G)-module.

Classical inequalities used in information theory such as those of de Bruijn, Fisher, Cramér, Rao, and Kullback carry over in a natural way from Euclidean space to unimodular Lie groups. These are groups that possess an integration measure that is simultaneously invariant under left and right shifts. All commutative groups are unimodular. And even in noncommutative cases unimodular Lie groups share many of the useful features of Euclidean space. The rotation and Euclidean motion groups, which are perhaps the most relevant Lie groups to problems in geometric mechanics, are unimodular, as are the unitary groups that play important roles in quantum computing. The extension of core information theoretic inequalities defined in the setting of Euclidean space to this broad class of Lie groups is potentially relevant to a number of problems relating to information gathering in mobile robotics, satellite attitude control, tomographic image reconstruction, biomolecular structure determination, and quantum information theory. In this paper, several definitions are extended from the Euclidean setting to that of Lie groups (including entropy and the Fisher information matrix), and inequalities analogous to those in classical information theory are derived and stated in the form of fifteen small theorems. In all such inequalities, addition of random variables is replaced with the group product, and the appropriate generalization of convolution of probability densities is employed. An example from the field of robotics demonstrates how several of these results can be applied to quantify the amount of information gained by pooling different sensory inputs.

Set inversion and box contraction on Lie groups using interval analysis