### A Contraction of S U (2) to the Heisenberg Group.

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It is an interesting open problem to establish Paley-Wiener theorems for general nilpotent Lie groups. The aim of this paper is to prove one such theorem for step two nilpotent Lie groups which is analogous to the Paley-Wiener theorem for the Heisenberg group proved in [4].

The group SU(1,d) acts naturally on the Hilbert space $L\xb2\left(Bd{\mu}_{\alpha}\right)(\alpha >-1)$, where B is the unit ball of ${\u2102}^{d}$ and $d{\mu}_{\alpha}$ the weighted measure ${(1-|z\left|\xb2\right)}^{\alpha}dm\left(z\right)$. It is proved that the irreducible decomposition of the space has finitely many discrete parts and a continuous part. Each discrete part corresponds to a zero of the generalized Harish-Chandra c-function in the lower half plane. The discrete parts are studied via invariant Cauchy-Riemann operators. The representations on the discrete parts are equivalent to actions on some holomorphic...

This paper is chiefly a survey of results obtained in recent years on the asymptotic behaviour of semigroups of bounded linear operators on a Banach space. From our general point of view, discrete families of operators ${T}^{n}:n=0,1,...$ on a Banach space X (discrete one-parameter semigroups), one-parameter ${C}_{0}$-semigroups $T\left(t\right):t\ge 0$ on X (strongly continuous one-parameter semigroups), are particular cases of representations of topological abelian semigroups. Namely, given a topological abelian semigroup S, a family of bounded...

We study a method of approximating representations of the group $M\left(n\right)$ by those of the group $SO(n+1)$. As a consequence we establish a version of a theorem of DeLeeuw for Fourier multipliers of ${L}^{p}$ that applies to the “restrictions” of a function on the dual of $M\left(n\right)$ to the dual of $SO(n+1)$.

A problem about representations of countable, commutative semigroups leads to an analytic non-Borel set.

Let S be a locally compact (σ-compact) group or semigroup, and let T(t) be a continuous representation of S by contractions in a Banach space X. For a regular probability μ on S, we study the convergence of the powers of the μ-average Ux = ʃ T(t)xdμ(t). Our main results for random walks on a group G are: (i) The following are equivalent for an adapted regular probability on G: μ is strictly aperiodic; ${U}^{n}$ converges weakly for every continuous unitary representation of G; U is weakly mixing for any...