Given a (reduced) locally compact quantum group A, we can consider the convolution algebra L(A) (which can be identified as the predual of the von Neumann algebra form of A). It is conjectured that L(A) is operator biprojective if and only if A is compact. The “only if” part always holds, and the “if” part holds for Kac algebras. We show that if the splitting morphism associated with L(A) being...

The aim of this paper is to show that the dynamics of L heat semigroups (p > 2) on a symmetric space of non-compact type is very different from the dynamics of the L heat semigroups if p ≤ 2. To see this, it is shown that certain shifts of the L heat semigroups have a chaotic behavior if p > 2 and that such a behavior is not possible in the cases p ≤ 2. These results are compared with the corre...

Generalizing the work of [5, 41], we give a general solution to the following problem: describe the triplets (Ω, g, μ) where g = (g(x)) is the (co)metric associated to the symmetric second order differential operator L(f) = 1 ρ ∑ ij ∂i(g ρ∂jf), defined on a domain Ω of R and such that L is expandable on a basis of orthogonal polynomials on L(μ), and dμ = ρ(x)dx is some admissible measure. Up to...

A new notion of SP-compactness is introduced in L-topological spaces by means of semi-preopen L-sets and their inequality, where L is a complete De Morgan algebra. This new notion does not depend on the structure of basis lattice L and L does not require any distributivity. This new notion implies semicompactness, hence it also implies compactness. This new notion is a good extension and it has...

We work in set-theory without choice ZF. Given a closed subset F of [0, 1] which is a bounded subset of l(I) (resp. such that F ⊆ l(I)), we show that the countable axiom of choice for finite subsets of I, (resp. the countable axiom of choiceACN) implies that F is compact. This enhances previous results where ACN (resp. the axiom of Dependent Choices DC) was required. Moreover, if I is linearly ...

Suppose / is a single-valued transformation of the metric space M onto the metric space fM. Many of the theorems of topology assert that if / is continuous, then it must have various other properties also. We deal here with the problem of determining under what circumstances certain of these properties are actually equivalent to continuity. The properties considered are: (1) fX is compact for e...

We study the following problem: describe triplets (?,g,?) where g=(g ij (x)) is (co)metric associated with symmetric second order differential operator L(f)=1 ?? ? i (g ?? j f) defined on a domain ? of ? d (that L diffusion reversible measure ?(dx)=?(x)dx) and such that there exists an orthonormal basis ? 2 (?) made polynomials which are at same time eigenvectors L, ranked according to their na...

In this paper, we investigate analytical and geometric properties of certain non-compact boundary-manifolds, namely manifolds of bounded geometry. One result are strong Bochner type vanishing results for the L-cohomology of these manifolds: if e.g. a manifold admits a metric of bounded geometry which outside a compact set has nonnegative Ricci curvature and nonnegative mean curvature (of the bo...