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1.
Let ( t ) t0 be a -semistable convolution semigroup of probability measures on a Lie groupG whose idempotent 0 is the Haar measure on some compact subgroupK. Then all the measures 1 are supported by theK-contraction groupC K() of the topological automorphism ofG. We prove here the structure theoremC K()=C()K, whereC() is the contraction group of . Then it turns out that it is sufficient to study semistable convolution semigroups on simply connected nilpotent Lie groups that have Lie algebras with a positive graduation.  相似文献   

2.
Several conditions are shown to be equivalent to the aperiodicity of a regular probability measure on a locally compact, separated topological GroupG. In particular, is aperiodic if and only if the sequence ( ( (n) denoting then-th convolution power of ) is convergent for any nonvoid open subsetU ofG with compact closure. It is always assumed that the support of generatesG as a closed semigroup.  相似文献   

3.
Let be a Guelfand measure (cf. [A, B]) on a locally compact groupG DenoteL 1 (G)=*L 1(G)* the commutative Banach algebra associated to . We show thatL 1 (G) is semi-simple and give a characterization of the closed ideals ofL 1 (G). Using the -spherical Fourier transform, we characterize all linear bounded operators inL 1 (G) which are invariants by -translations (i.e. such that 1(( x f) )=( x ((f)) for eachxG andfL 1 (G); where x f(y)=f(xy); x,y G). WhenG is compact, we study the algebraL 1 (G) and obtain results analogous to ones obtained for the commutative case: we show thatL 1 (G) is regular, all closed sets of its Guelfand spectrum are sets of synthesis and establish theorems of harmonic synthesis for functions inL p (G) (p=1,2 or +).
  相似文献   

4.
Let be a probability measure generating a locally compact semigroup S. If the convolution sequence n is tight, in particular if S is compact, S admits a closed minimal ideal K. The convergence of n is characterized in terms of convergence of a homomorphic image (~) n on a factor group of the compact group G in the Rees–Suschkewitsch decomposition of K.  相似文献   

5.
Summary Consider a random walk of law on a locally compact second countable groupG. Let the starting measure be equivalent to the Haar measure and denote byQ the corresponding Markov measure on the space of pathsG . We study the relation between the spacesL (G , a ,Q) andL (G , i ,Q) where a and i stand for the asymptotic and invariant -algebras, respectively. We obtain a factorizationL (G , a ,Q) L (G , i ,Q)L (C) whereC is a cyclic group whose order (finite or infinite) coincides with the period of the Markov shift and is determined by the asymptotic behaviour of the convolution powers n.  相似文献   

6.
Given a sequence of probability measures ( n ) on a finite abelian semigroup, we present necessary and sufficient conditions which guarantee the weak convergence of the convolution products k,n k+1*···* n (k<n), asn for allk0. These conditions are verifiable in the sense that they are based entirely on the individual measures in the sequence ( n ).  相似文献   

7.
LetG be a stratified Lie group and (t)t 0 be a continuous convolution semigroup of probability measures onG. A probability measurev is said to belong to the -domain of attraction of 1, if there exists a sequence (a n ) of positive real numbers such that weakly, where 1 denotes the natural dilation onG. We prove convergence criteria for discrete convolution semigroups. These are used to obtain a simple necessary and sufficient condition for the existence of sucha n if (t)t 0 has no Gaussian component. For the proof we introduce the notion of regularly varying measures onG and develop the necessary theory of regular variation.  相似文献   

8.
A locally compact group G is called a Tortrat group if for any probability measure on G which is not idempotent, the closure of {gg –1 | gG} does not contain any idempotent measure. We show that a connected Lie group G is a Tortrat group if and only if for all gG all eigenvalues of Ad g are of absolute value 1. Together with well-known results this also implies that a connected locally compact group is a Tortrat group if and only if it is of polynomial growth.  相似文献   

9.
Given a probability measure, , on a locally compact group, necessary and sufficient support conditions which ensure that the concentration functions associated with converge to zero have previously been determined. In this note the rate of this convergence when is adapted on a discrete group G is shown to depend on the volume growth rate of N , the smallest normal subgroup a coset of which contains the support of .  相似文献   

10.
LetS be a locally compact semigroup. It is shown that if a measure is absolutely continuous and ifS is cancellative, then the measure concentrated on a Borel subsetB ofS (i. e. =(B.)) is also absolutely continuous. Other properties of absolutely continuous measures will be obtained. Moreover we will answer the question when absolutely continuous probability measures exist. This is the case ifS admits an invariant integral on the space of all continuous functions onS with compact support. Another result is the following: If the compact semigroupS has a connected kernel then there exist absolutely continuous probability measures if and only ifS is amenable.  相似文献   

11.
For any locally compact groupG, we show that any locally tight homomorphism from a real directed semigroup intoM 1 (G) (semigroup of probability measures onG) has a shift which extends to a continuous one-parameter semigroup. IfG is ap-adic algebraic group then the above holds even iff is not locally tight. These results are applied to give sufficient conditions for embeddability of some translate of limits of sequences of the form {v n kn } and M 1 (G) such that ()= M , for somek>1 and AutG (cf. Theorems 2.1, 2.4, 3.7).  相似文献   

12.
We obtain several homotopy obstructions to the existence of non-closed connected Lie subgroupsH in a connected Lie groupG.First we show that the foliationF(G, H) onG determined byH is transversely complete [4]; moreover, forK the closure ofH inG, F(K, H) is an abelian Lie foliation [2].Then we prove that 1(K) and 1(H) have the same torsion subgroup, n (K)= n (H) for alln 2, and rank1(K) — rank1(H) > codimF(K, H). This implies, for instance, that a contractible (e.g. simply connected solvable) Lie subgroup of a compact Lie group must be abelian. Also, if rank1(G) 1 then any connected invariant Lie subgroup ofG is closed; this generalizes a well-known theorem of Mal'cev [3] for simply connected Lie groups.Finally, we show that the results of Van Est on (CA) Lie groups [6], [7] provide many interesting examples of such foliations. Actually, any Lie group with non-compact centre is the (dense) leaf of a foliation defined by a closed 1-form. Conversely, when the centre is compact, the latter is true only for (CA) Lie groups (e.g. nilpotent or semisimple).  相似文献   

13.
Perturbation of Dirichlet forms by measures   总被引:8,自引:0,他引:8  
Perturbations of a Dirichlet form by measures are studied. The perturbed form –++ is defined for in a suitable Kato class and + absolutely continuous with respect to capacity. L p-properties of the corresponding semigroups are derived by approximating by functions. For treating +, a criterion for domination of positive semigroups is proved. If the unperturbed semigroup has L p -L q -smoothing properties the same is shown to hold for the perturbed semigroup. If the unperturbed semigroup is holomorphic on L 1 the same is shown to be true for the perturbed semigroup, for a large class of measures.  相似文献   

14.
Consider a closed subgroup of the automorphism group of a homogeneous treeT, and assume that acts transitively on the vertex set. Suppose that is a probability measure on which has continuous density with respect to Haar measure and whose support is compact open and generates as a closed semigroup. It is shown that the Martin boundary of with respect to the random walk with law coincides with the space of ends ofT. This extends known results for free groups and applies, for example, to the affine group over a non archimedean local field.  相似文献   

15.
LetT be an eight-dimensional, connected, locally compact ternary field and let denote a connected closed Lie subgroup of its automorphism group which is taken with the compact-open topology. It is proved that if the ternary fixed fieldF of is connected, then is either isomorphic to one of the compact Lie groupsG 2 or SU3, or the (covering) dimension of is at most 7.  相似文献   

16.
A word w is said to be a primitive word if it cannot be expressed as a power of any other word. A language L consisting of non-empty words is called -reducible if there exists a non-empty word w such that Lw contains only finitely many powers of each primitive word. We show that every regular component, context-free component, local language and every regular language containing no primitive words are -reducible. Languages which are not -reducible are investigated and characterized. We show that every code is -reducible. But there are 2-codes which are not -reducible. The -annihilator of a language L is the set of all non-empty words w such that Lw contains only finitely many powers of each primitive word. This paper also concerns the properties of the -annihilators of languages. The -annihilators of 2-codes and some other languages are investigated and characterized in this paper. The results provide an outline of the relationship between the catenation of languages and the powers of primitive words.  相似文献   

17.
The concepts of class L measures and selfdecomposable measures are generalised from vector spaces to simply connected nilpotent groups G. It has been shown that any full class L (probability) measure on G can be decomposed as = 1* ... * n, where each i is a selfdecomposable measure on a subgroup G i; itself is selfdecomposable under certain additional conditions—for example, when is symmetric. This generalizes a well known result on vector spaces. Some examples of class L measures on G are also constructed.  相似文献   

18.
Let be a probability measure on a locally compact groupG. A real Borel functionf onG is called -harmonic if it satisfies the convolution equation *f=f. Given that isnonsingular with its translates, we show that the bounded -harmonic functions are constant on a class of groups including the almost connected [IN]-groups. If is nondegenerate and absolutely continuous, we solve the more general equation *= for positive measure on those groups which are metrizable and separable.Supported by Hong Kong RGC Earmarked Grant and CUHK Direct Grant  相似文献   

19.
It is shown, that for the action of a -compact group, being amenable as an abstract discrete group, on a locally compact measure space (X, , ), is not the unique invariant mean. Furthermore, this paper gives a characterisation of probability spaces, having a unique invariant mean for the action of an amenable group.  相似文献   

20.
The fundamental result: if and v are two finite Borel measures, defined in the spaceL p[0, 1] (1p<) or in C(K) (K is a metric compactum without isolated points), then from the equalities (B)=v(B) for all balls B of radius 1 there follows that =v. In addition, in the spaces C(K) and p (1p<) from the inequalities (B) v(B) there follows that v.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 177, pp. 122–128, 1989.  相似文献   

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