Related papers: Continuity of convolution of test functions on Lie…

Let $G$ be a non-compact group, $K$ the compact subgroup fixed by a Cartan involution and assume $G/K$ is an exceptional, symmetric space, one of Cartan type $E,F $ or $G$. We find the minimal integer, $L(G),$ such that any convolution…

We prove a criterion for continuity of bilinear maps on countable direct sums of topological vector spaces. As a first application, we get a new proof for the fact (due to Hirai et al. 2001) that the map taking a pair of test functions on…

We consider convolution equations of the type f * T = g where f, g are in L^p(R^n) and T is a compactly supported distribution. Under natural assumptions on the zero set of the Fourier transform of T we show that f is compactly supported,…

Let $G$ be a locally compact group and also let $H$ be a compact subgroup of $G$. It is shown that, if $\mu$ is a relatively invariant measure on $G/H$ then there is a well-defined convolution on $L^1(G/H,\mu)$ such that the Banach space…

Consider the self-map F of the space of real-valued test functions on the line which takes a test function f to the test function sending a real number x to f(f(x))-f(0). We show that F is discontinuous, although its restriction to the…

Let the measure algebra of a topological group be equipped with the topology of uniform convergence on bounded right uniformly equicontinuous sets of functions. Convolution is separately continuous on the measure algebra, and it is jointly…

Let $G$ be a compact Lie group of Lie type $B_{n},$ such as $SO(2n+1)$. We characterize the tuples\ $(x_{1},...,x_{L})$ of the elements $x_{j}\in G$ which have the property that the product of their conjugacy classes has non-empty interior.…

Let G be a Lie group modelled on a locally convex space, with Lie algebra g, and k be a non-negative integer or infinity. We say that G is C^k-semiregular if each C^k-curve c in g admits a left evolution Evol(c) in G. If, moreover, the map…

We study (weakly) continuous convolution semigroups of probability measures on a Lie group G or a homogeneous space G/K, where K is a compact subgroup. We show that such a convolution semigroup is the convolution product of its initial…

We conjecture that if $G$ is a simple compact Lie group with trivial center, then every $d$-variable non-constant word map with coefficients in $G$ defines a non-constant function on $G^d$. We prove the conjecture for $A_r$, $B_r$, $E_6$,…

A locally compact contraction group is a pair (G,f) where G is a locally compact group and f an automorphism of G which is contractive in the sense that the forward orbit under f of each g in G converges to the neutral element e, as n tends…

We construct a universal continuous invariant bilinear form for the Lie algebra of compactly supported sections of a Lie algebra bundle in a topological sense. Moreover we construct a universal continuous central extension of a current…

Let G be a Lie group and E be a locally convex topological G-module. If E is sequentially complete, then E and its space of smooth vectors are modules for the algebra D(G) of compactly supported smooth functions on G. However, the module…

If E is a locally convex topological vector space, let P(E) be the pre-ordered set of all continuous seminorms on E. We study, on the one hand, for g an infinite cardinal those locally convex spaces E which have the g-neighbourhood property…

Let $\mathfrak{g}$ be a compact, simple Lie algebra of dimension $d$. It is a classical result that the convolution of any $d$ non-trivial, $G$ -invariant, orbital measures is absolutely continuous with respect to Lebesgue measure on…

Let $M$ be a compact, real analytic manifold and $G$ be the Lie group of all real-analytic diffeomorphisms of $M$, which is modelled on the space ${\mathfrak g}$ of real-analytic vector fields on $M$. We study flows of time-dependent…

We solve the regularity problem for Milnor's infinite dimensional Lie groups in the $C^0$-topological context, and provide necessary and sufficient regularity conditions for the (standard) $C^k$-topological setting. We prove that the…

Let $G/K$ be an irreducible symmetric space where $G$ is a non-compact, connected Lie group and $K$ is a compact, connected subgroup. We use decay properties of the spherical functions to show that the convolution product of any $r=r(G/K)$…

Convolution semigroups of states on a quantum group form the natural noncommutative analogue of convolution semigroups of probability measures on a locally compact group. Here we initiate a theory of weakly continuous convolution semigroups…

Let $C(X,E)$ be the linear space of all continuous functions on a compact Hausdorff space $X$ with values in a locally convex space $E$. We characterize maps $T:C(X,E)\to C(Y,E)$ which satisfy $\mathrm{Ran}(TF-TG)\subset\mathrm{Ran}(F-G)$…