In this paper sufficient conditions are given in order that every distribution invariant under a Lie group extend from the set of orbits of maximal dimension to the whole of the space. It is shown that these conditions are satisfied for the n-point action of the pure Lorentz group and for a standard action of the Lorentz group of arbitrary signature.

The convolution of ultrafilters of closed subsets of a normal topological group is considered as a substitute of the extension onto ${\left(\beta \right)}^2$ of the group operation. We find a subclass of ultrafilters for which this extension is well-defined and give some examples of pathologies. Next, for a given locally compact group and its dense subgroup , we construct subsets of β algebraically isomorphic to . Finally, we check whether the natural mapping from β onto β is a homomorphism with respect to the extension...

We prove that the Wyler completion of the unitary Cauchy space on a given Hausdorff topological 5 monoid consisting of the underlying set of this monoid and of the family of unitary Cauchy filters on it, is a T2-topological space and, in the commutative case, an abstract monoid containing the initial one.

Elements of the general theory of Lie-Cartan pseudogroups (including the intransitive case) are developed within the framework of infinitely prolonged systems of partial differential equations (diffieties) which makes it independent of any particular realizations by transformations of geometric object. Three axiomatic approaches, the concepts of essential invariant, subgroup, normal subgroup and factorgroups are discussed. The existence of a very special canonical composition series based on Cauchy...

We give new and simple sufficient conditions for Gaussian upper bounds for a convolution semigroup on a unimodular locally compact group. These conditions involve certain semigroup estimates in L²(G). We describe an application for estimates of heat kernels of complex subelliptic operators on unimodular Lie groups.

[For the entire collection see Zbl 0742.00067.]Let ${𝔤}_k$ be the Lie algebra $𝔤l(k,𝒞)$, and let $U_k$ be the universal enveloping algebra for ${𝔤}_k$. Let $Z_k$ be the center of $U_k$. The authors consider the chain of Lie algebras ${𝔤}_n\supset {𝔤}_{n-1}\supset \cdots \supset {𝔤}_1$. Then $Z=\langle Z_k\mid k=1,2,\cdots n\rangle$ is an associative algebra which is called the Gel’fand-Zetlin subalgebra of $U_n$. A ${𝔤}_n$ module $V$ is called a $GZ$-module if $V={\sum }_x\oplus V\left(x\right)$, where the summation is over the space of characters of $Z$ and $V\left(x\right)=\{v\in V\mid {(a-x\left(a\right))}^{m}v=0$, $m\in {𝒵}_{+}$, $a\in 𝒵\}$. The authors describe several properties of $GZ$- modules. For example, they prove that if $V\left(x\right)=0$ for some $x$...

Let G be a locally compact group. Let σ be a continuous involution of G and let μ be a complex bounded measure. In this paper we study the generalized d'Alembert functional equationD(μ)    ∫G f(xty)dμ(t) + ∫G f(xtσ(y))dμ(t) = 2f(x)f(y) x, y ∈ G;where f: G → C to be determined is a measurable and essentially bounded function.

Let $\tau$ be a continuous unitary representation of the locally compact group $G$ on the Hilbert space $H_{\tau }$. Let $C_{\tau }^{*}[V{N}_{\tau }]$ be the $C^{*}[W^{*}]$ algebra generated by$$(L^1\left(G\right))\text{and}{M}_{\tau }(C_{\tau }^{*})=\left\{\phi \in V{N}_{\tau };\phi C_{\tau }^{*}+C_{\tau }^{*}\phi \subset C_{\tau }^{*}\right\}.$$The main result obtained in this paper is Theorem 1:If $G$ is $\sigma$-compact and $M_{\tau }(C_{\tau }^{*})=V{N}_{\tau }$ then supp $\tau$ is discrete and each $\pi$ in supp $\tau$ in CCR.We apply this theorem to the quasiregular representation $\tau ={\pi }_H$ and obtain among other results that $M_{{\pi }_H}(C_{{\pi }_H}^{*})=V{N}_{{\pi }_H}$ implies in many cases that $G/H$ is a compact coset space.