B. Kitchens and K. Schmidt studied algebraic Z d -actions on compact abelian groups. Those actions arise as duals of discrete modules. Motivated by Schmidt's book we consider in this paper actions of more general groups, in particular of the discrete Heisenberg group. We generalize the algebraic characterizations of dynamical properties. Mathematics Subject Classification (2000). 37A25, 22D40.The main purpose of this paper is to generalize results of the theory for algebraic Z d -actions, which is developed in a series of papers starting with [4] and is presented in K. Schmidt's book [8], to actions of non-abelian groups. The discrete Heisenberg group turns out to be a significant example, demonstrating the problems in this generalized setting. We give a very general characterization of ergodicity for such actions. As we will see, the situation is different for mixing properties. We also state some results concerning entropy and expansiveness. In the last section we examine a number of different actions of the Heisenberg group exhibiting the analogies and differences to the abelian case (i.e. to Z d -actions).
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Abstract. Let Sl(G) be a Segal algebra on a locally compact group. The central functions of S '(G) are dense in the center of L\G). S\G) has central approximate units iff G G [SIN]. This is a generalization of a result of Reiter on the one hand and of Mosak on the other hand. The proofs depend on the structure theorems of [S/JV]-and [W]-groups. In the second part some new examples of Segal algebras are constructed. A locally compact group is discrete or Abelian iff every Segal algebra is rightinvariant. As opposed to the results, the proofs are not quite obvious.
Abstract.We prove, that two concepts of weak containment do not coincide, contradicting results in [1, Lemma 3.3 and Proposition 3.4]. The statement of Theorem 3.5 remains valid. There exist infinite tall compact groups G (i.e. the set {a e G, dim er = n} is finite for each positive integer n ) having the mean-zero weak containment property. Such groups do not have the dual Bohr approximation property or AP(G) ^ Cg(G).Let G be a compact group, then G is said to have the mean-zero weak containment property if there exists a net {ga} in LliG) = {feL2(G),Jf(x)dx = 0 such that ||gQ||2 = 1 and limQ \\Lyga -ga\\2 = 0 for all y e G. Ly denotes the left translation operator, Lyfix) = f{y~xx), see [4]. It was claimed in [1, Lemma 3.3], that this property were equivalent to the following weaker version:There exists a net ha e L\(G), \\ha\\2 = 1, such that Lyha -ha -* 0 weakly in L¡iG).The statement of this lemma is false. We show, why the indicated proof does not work (the error in the proof of Lemma 3.3 [1] is not isolated in literature). Since for convex sets the weak closure coincides with the norm closure, we can replace (ha) by some convex linear combinations ga suchthat lim||Lj,£0-ga\\2 = 0, Vy e G. But unfortunately it may happen, that lim ||ga||2 = 0.In fact, if G is an infinite compact group, then L^(G) is infinite dimensional, therefore there exists a net ha e L^(G) , such that \\ha\\2 = 1 and lima(ha, f) = 0 for all f e LliG), consequently lixn(Lyha -ha , f) = lixn(ha, Ly^xf)-lixn(ha,f) = 0 a a a for all / e L¡(G).
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