Invariant means on the continuous bounded functions

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 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

In this paper, we introduce new notions of $I$-biflatness and $I$-biprojectivity, for a Banach algebra $A$, where $I$ is a closed ideal of $A$. We show that $M(G)$ is $L^{1}(G)$-biprojective ($I$-biflat) if and only if $G$ is a compact group (an amenable group), respectively. Also, we show that, for a non-zero ideal $I$, if the Fourier algebra $A(G)$ is $I$-biprojective, then $G$ is a discrete group. Some examples are given to show the differences between these new notions and the classical ones.

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 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 $G$ be a Lie group, and let $D$ be a discrete subgroup of $G$ such that the right coset space $D\backslash G$ has finite right-invariant volume. We will exhibit an injection of left-invariant means on ${l^\infty }(D)$ into left-invariant means on the left uniformly continuous bounded functions of $G$. When $G$ is an abelian Lie group with finitely many connected components, we also show surjectivity, and when $G$ is the additive group ${{\mathbf {R}}^n}$ and $D$ is ${{\mathbf {Z}}^n}$, the bijection will explicitly take the form of an integral over the unit cube ${[0,1]^n}$.

This article concerns the relationship between the spectral and the norm properties of certain subalgebras of the convolution measure algebra M(G) (Borel regular finite measures) on a locally compact amenable group G.

At a fixed point x? E Rn the (real) zeroes of Pm(xO, 6) form a cone in Rn which is called the (real) characteristic cone of P(x, D) at x?. In this paper we study only differential operators with flat characteristic cones. Specifically we assume that at each point of an open set Q the characteristic cone of P(x, D) is contained in the orthogonal complement of a fixed subspace W of Rn different from {O}. In ?2, by a repeated application of llolmgren's theorem (as extended to distribution solutions by Hormander [1]), we show that if P(x, D) has analytic coefficients in Q and if u is a distribution solution of P(x, D)u = 0 in Q which vanishes in an open subset QO of Q, then u must also vanish in the Wextension of QO in Q consisting of all points of Q which can be connected to points of QO by polygonal paths in Q with sides defining vectors in W. We consider next the problem of characterizing geometrically the open sets in Rn which are P-convex with respect to a differential operator P(D) with constant coefficients. An open set Q in Rn is called P-convex if to every compact subset K1 of Q there exists a compact subset K2 of Q such that for every u E &'(Q) (distribution with compact support in Q),

Building on our previous work, we study the non-relative homology of quantum group convolution algebras. Our main result establishes the equivalence of amenability of a locally compact quantum group and 1-injectivity of as an operator -module. In particular, a locally compact group G is amenable if and only if its group von Neumann algebra VN(G) is 1-injective as an operator module over the Fourier algebra A(G). As an application, we provide a decomposability result for completely bounded -module maps on , and give a simpliûed proof that amenable discrete quantum groups have co-amenable compact duals, which avoids the use of modular theory and the Powers-Størmer inequality, suggesting that our homological techniques may yield a new approach to the open problem of duality between amenability and co-amenability.

The lemma on matrices, proved in this paper, replaces, in many instances, the Baire category arguments. As simple consequences of the lemma we get Orlicz-Pettis and Nikidym type theorems.

Let G G be a locally compact group P ( G ) = { 0 ≦ ϕ ∈ L 1 ( G ) ; ∫ ϕ ( x ) d x = 1 } P(G) = \{ 0 \leqq \phi \in {L_1}(G);\int {\phi (x)dx = 1\} } and ( l a f ) ( x ) = a f ( x ) = f ( a x ) ({l_a}f)(x) = {}_af(x) = f(ax) for all a , x ∈ G a,x \in G and f ∈ L ∞ ( G ) .0 ≦ Ψ ∈ L ∞ ( G ) ∗ , Ψ ( 1 ) = 1 f \in {L^\infty }(G).0 \leqq \Psi \in {L^\infty }{(G)^ \ast },\Psi (1) = 1 is said to be a [topological] left invariant mean ([TLIM] LIM) if Ψ ( a f ) = Ψ ( f ) [ Ψ ( ϕ ∗ f ) = Ψ ( f ) \Psi {{\text {(}}_a}f) = \Psi (f)[\Psi (\phi \ast f) = \Psi (f) ] for all a ∈ G , ϕ ∈ P ( G ) , f ∈ L ∞ ( G ) a \in G,\phi \in P(G),f \in {L^\infty }(G) . The main result of this paper is the Theorem. Let G G be a locally compact group, amenable as a discrete group. If G G contains an open σ \sigma -compact normal subgroup, then LIM = TLIM if and only if G G is discrete. In particular if G G is an infinite compact amenable as discrete group then there exists some Ψ ∈ L I M \Psi \in LIM which is different from normalized Haar measure. A harmonic analysis type interpretation of this and related results are given at the end of this paper . 2 ^{2}

This thesis is dedicated to studying the theory of entropy and its relation to the Kolmogorov complexity. Originating in physics, the notion of entropy was introduced to mathematics by C. E. Shannon as a way of measuring the rate at which information is coming from a data source. There are, however, a few different ways of telling how much information there is: an alternative approach to quantifying the amount of information is the Kolmogorov complexity, which was proposed by A. N. Kolmogorov. The Shannon entropy is the key ingredient in the definition of the Kolmogorov-Sinai entropy of a measure-preserving systems. Roughly speaking, the Kolmogorov-Sinai entropy is the expected amount of information in `Shannon sense' that one obtains per unit of time by observing the evolution of a measure-preserving system. In topological dynamics, the topological entropy takes place of the Kolmogorov-Sinai entropy. For metrizable systems, the topological entropy measures the exponential growth rate of the number of distinguishable partial orbits of length n as n tends to infinity. Originally defined for Z-actions, the 'classical' theories of entropy were later extended to actions of amenable groups. We provide a necessary background on amenable groups, topological/measurepreserving dynamics and the entropy theory in Chapters 1, 2, 3 and 5. The main focus of this thesis is extending the following results. First of all, a common generalization of the topological and the Kolmogorov-Sinai entropy theories for Z-systems was suggested by G. Palm. We provide an abstract generalization of the work of Palm for actions of discrete amenable groups in the language of measurement functors in Chapter 6. Secondly, we investigate the connection of entropy and Kolmogorov complexity. Originally, the equality between the topological entropy and a certain quantity measuring maximal asymptotic Kolmogorov complexity of the trajectories was established by A. A. Brudno for subshifts over Z. Later, he proved the equality of the Kolmogorov-Sinai entropy and the asymptotic Kolmogorov complexity of almost every trajectory for ergodic subshifts over Z. We provide a generalization of these results as follows. Firstly, in Chapter 4 we give a background on computability and Kolmogorov complexity and, further, introduce computable Folner monotilings, which are central in our extensions of Brudno's results. We treat the 'first' and the 'second' theorems of Brudno in Chapter 7. The first theorem is generalized for subshifts over computable groups admitting computable Folner monotilings, while the second theorem is proved under the assertion of regularity of the monotiling, which we introduce in Chapter 7 as well.

Hardy’s uncertainty principle for the Gabor transform is proved for locally compact abelian groups having noncompact identity component and groups of the form$\mathbb{R}^{n}\times K$, where$K$is a compact group having irreducible representations of bounded dimension. We also show that Hardy’s theorem fails for a connected nilpotent Lie group$G$which admits a square integrable irreducible representation. Further, a similar conclusion is made for groups of the form$G\times D$, where$D$is a discrete group.

In the sequel ofB. E. Johnson's work on amenable Banach algebras we characterize amenable and compact groups.

Given a locally compact group $G$ and a unitary representation $ρ:G\to U({\mathcal H})$ on a Hilbert space ${\mathcal H}$, we construct a $C^*$-correspondence ${\mathcal E}(ρ)={\mathcal H}\otimes_{\mathbb C} C^*(G)$ over $C^*(G)$ and study the Cuntz-Pimsner algebra ${\mathcal O}_{{\mathcal E}(ρ)}$. We prove that for $G$ compact, ${\mathcal O}_{{\mathcal E}(ρ)}$ is strong Morita equivalent to a graph $C^*$-algebra. If $λ$ is the left regular representation of an infinite, discrete and amenable group $G$, we show that ${\mathcal O}_{{\mathcal E}(λ)}$ is simple and purely infinite, with the same $K$-theory as $C^*(G)$. If $G$ is compact abelian, any representation decomposes into characters and determines a skew product graph. We illustrate with several examples and we compare ${\mathcal E}(ρ)$ with the crossed product $C^*$-correspondence.