An invariant mean on a group G is a normalized, positive, translation invariant linear functional defined on the space of all bounded complex valued functions on G. Some groups possess an invariant mean (or are said to be amenable), while others do not. In particular, all abelian groups are amenable [2, §17.5]. An invariant mean on a group need not be inversion invariant. It is quite easy to sh...

For a locally compact group G, we prove that a topological inner invariant mean on LUC(G) has an extension to a topological inner invariant mean on L∞(G). Mathematics Subject Classification: 43A07, 43A10

We propose a definition of a “C-Eberlein” algebra, which is a weak form of a C-bialgebra with a sort of “unitary generator”. Our definition is motivated to ensure that commutative examples arise exactly from semigroups of contractions on a Hilbert space, as extensively studied recently by Spronk and Stokke. The terminology arises as the Eberlein algebra, the uniform closure of the Fourier-Stiel...

Let X1 and X2 be two independent random variables from gamma populations Pi1,P2 with means alphaθ1 and alphaθ2 respectively, where alpha(> 0) is the common known shape parameter and θ1 and θ2 are scale parameters. Let X(1) ≤ X(2) denote the order statistics ofX1 and X2. Suppose that the population corresponding to the largest X(2) (or the smallest X(1)) observation is selected. The problem ofin...

Applying Šmulian Theorem, we show that if a Banach space X is not reflexive then the space of bounded functions from Z with values in X does not admit an invariant mean. Invariant means on amenable groups are an important tool in many parts of mathematics, especially in harmonic analysis (see [5, 6]). For basic properties of invariant means, we refer the reader to [5]. We would only like to men...

We show that topological amenability of an action of a countable discrete group on a compact space is equivalent to the existence of an invariant mean for the action. We prove also that this is equivalent to vanishing of bounded cohomology for a class of Banach G-modules associated to the action, as well as to vanishing of a specific cohomology class. In the case when the compact space is a poi...