For $\varphi$ a normalized positive definite function on a locally compact abelian group $G$, let ${\pi }_{\varphi }$ be the unitary representation associated to $\varphi$ by the GNS construction. We give necessary and sufficient conditions for the vanishing of 1-cohomology $H^1(G,{\pi }_{\varphi })$ and reduced 1-cohomology ${\overline{H}}^1(G,{\pi }_{\varphi })$. For example, ${\overline{H}}^1(G,{\pi }_{\varphi })=0$ if and only if either $\text{Hom}(G,ℂ)=0$ or ${\mu }_{\varphi }\left(1_G\right)=0$, where $1_G$ is the trivial character of $G$ and ${\mu }_{\varphi }$ is the probability measure on the Pontryagin dual $\widehat{G}$ associated to $\varphi$ by Bochner’s Theorem. This streamlines an argument of Guichardet (see Theorem...

The Banach spaces $\Lambda (A,B,X,G)$ defined in this paper consist essentially of those elements of $L^1\left(G\right)$ ($G$ being a locally compact group) which can in a certain sense be well approximated by functions with compact support. The main result of this paper is the fact that in many cases $\Lambda (A,B,X,G)$ becomes a Banach convolution algebra. There exist many natural examples. Furthermore some theorems concerning inclusion results and the structure of these spaces are given. In particular we prove that simple conditions imply the existence...

Let (G,τ) be a Hausdorff Abelian topological group. It is called an s-group (resp. a bs-group) if there is a set S of sequences in G such that τ is the finest Hausdorff (resp. precompact) group topology on G in which every sequence of S converges to zero. Characterizations of Abelian s- and bs-groups are given. If (G,τ) is a maximally almost periodic (MAP) Abelian s-group, then its Pontryagin dual group ${(G,\tau )}^{\wedge }$ is a dense -closed subgroup of the compact group ${\left(G_d\right)}^{\wedge }$, where $G_d$ is the group G with the discrete...

We present a new approach to determining supports of extreme, normed by 1, positive definite class functions of discrete groups, i.e. characters in the sense of E. Thoma [8]. Any character of a group produces a unitary representation and thus a von Neumann algebra of linear operators with finite normal trace. We use a theorem of H. Umegaki [9] on the uniqueness of conditional expectation in finite von Neumann algebras. Some applications and examples are given.

We study the densities of the semigroup generated by the operator $-X^2+\left|Y\right|$ on the 3-dimensional Heisenberg group. We show that the 7th derivatives of the densities have a jump discontinuity. Outside the plane x=0 the densities are $C^{\infty }$. We give explicit spectral decomposition of images of $-X^2+\left|Y\right|$ in representations.

If G is a locally compact group with a compact invariant neighbourhood of the identity e, the following property (*) holds: For every continuous positive definite function h≥ 0 with compact support there is a constant $C_h>0$ such that $\int L_{x}h·g\le C_h\int hg$ for every continuous positive definite g≥0, where $L_x$ is left translation by x. In [L], property (*) was stated, but the above inequality was proved for special h only. That “for one h” implies “for all h” seemed obvious, but turned out not to be obvious at all. We fill...