For locally compact groups G, Kuan Yuan studied a notion of inner amenability groups, that is, if there exists an inner invariant mean on G. In this article, among other things, the author investigates the inner amena...For locally compact groups G, Kuan Yuan studied a notion of inner amenability groups, that is, if there exists an inner invariant mean on G. In this article, among other things, the author investigates the inner amenability on a locally compact group G. The author gives sufficient conditions and some necessary conditions about G to have an inner invariant mean.展开更多
Let G be a locally compact group and let F be a closed subgroup of G × G. Pier introduced the notion of F-amenability which gives a new classification of groups. This concept generalizes the concept of amenabilit...Let G be a locally compact group and let F be a closed subgroup of G × G. Pier introduced the notion of F-amenability which gives a new classification of groups. This concept generalizes the concept of amenability and inner amenability for locally compact groups. In this paper, among other things, we extend some standard results for amenable groups to F-amenable groups and give various characterizations for F-amenable groups. A sequence of characterizations of F-amenable groups is given here by developing the well-known Flner's conditions for amenable locally compact groups. Several characterizations of inner amenability are also given.展开更多
In this note, we consider the multipliers on weighted function spaces over totally disconnected locally compact abelian groups (Vilenkin groups). Firstly we show an (H1 ,L ) multiplier result. We also give an (Hap ,Ha...In this note, we consider the multipliers on weighted function spaces over totally disconnected locally compact abelian groups (Vilenkin groups). Firstly we show an (H1 ,L ) multiplier result. We also give an (Hap ,Hap) multiplier result under a similiar condition of Lu Yang type. In section 2, we obtain a result about the boundedness of multipliers on weighted Besov spaces.展开更多
Three classical compactification procedures are presented with nonstandard flavour. This is to illustrate the applicability of Nonstandard analytic tool to beginners interested in Nonstandard analytic methods. The gen...Three classical compactification procedures are presented with nonstandard flavour. This is to illustrate the applicability of Nonstandard analytic tool to beginners interested in Nonstandard analytic methods. The general procedure is as follows: A suitable equivalence relation is defined on an enlargement <sup>*</sup><em>X </em>of the space <em>X</em> which is a completely regular space or a locally compact Hausdorff space or a locally compact Abelian group. Accordingly, every <em>f</em> in <em>C</em>(<em>X</em>,<em>R</em>) (the space of bounded continuous real valued functions on <em>X</em>) or <em>Cc</em>(<em>X</em>,<em>R</em>) (the space of continuous real valued functions on <em>X</em> with compact support) or the dual group <span style="white-space:nowrap;">Γ </span>of the locally compact Abelian group <em>G</em> is extended to the set <img alt="" src="Edit_b9535172-924d-44f0-bab3-c49db17a3b7a.png" /> of the above mentioned equivalence classes. A compact topology on <img alt="" src="Edit_9d7962a3-b8a3-4693-b62a-078c8c4b4853.png" /> is obtained as the weak topology generated by these extensions of <em>f</em>. Then <em>X</em> is naturally imbedded densely in <img alt="" src="Edit_f7d403b2-eff3-4555-b8e7-1b106e06d2e7.png" />.展开更多
Two non-discrete Hausdorff group topologiesτandδon a group G are called transversal if the least upper boundτ⋁δofτandδis the discrete topology.In this paper,we discuss the existence of transversal group topologi...Two non-discrete Hausdorff group topologiesτandδon a group G are called transversal if the least upper boundτ⋁δofτandδis the discrete topology.In this paper,we discuss the existence of transversal group topologies on locally pseudocompact,locally precompact,or locally compact groups.We prove that each locally pseudocompact,connected topological group satisfies central subgroup paradigm,which gives an affirmative answer to a problem posed by Dikranjan,Tkachenko,and Yaschenko[Topology Appl.,2006,153:3338–3354].For a compact normal subgroup K of a locally compact totally disconnected group G,if G admits a transversal group topology,then G/K admits a transversal group topology,which gives a partial answer again to a problem posed by Dikranjan,Tkachenko,and Yaschenko in 2006.Moreover,we characterize some classes of locally compact groups that admit transversal group topologies.展开更多
Let π be a unitary representation of a locally compact topological group G on a separable Hilbert space H. A vector ψ∈ H is called a continuous frame wavelet if there exist A, B 〉 0 such that A||Ф||2≤fG|...Let π be a unitary representation of a locally compact topological group G on a separable Hilbert space H. A vector ψ∈ H is called a continuous frame wavelet if there exist A, B 〉 0 such that A||Ф||2≤fG|〈π(G)ψ,Ф|2dg≤B||Ф||2(Ф∈H),in which dg is the left Haar measure of G. Similar to the study of wavelets, an essential problem in the study of continuous frame wavelets is how to characterize them under the given unitary representation. Moreover, we investigate a relation between admissible vectors of π and its components.展开更多
In this paper we study the solutions and stability of the generalized Wilson's functional equation fc f(xty)dtt(t) + fc f(xtσ(y))dtt(t) =2f(x)g(y), x,y C G, where G is a locally compact group, a is a ...In this paper we study the solutions and stability of the generalized Wilson's functional equation fc f(xty)dtt(t) + fc f(xtσ(y))dtt(t) =2f(x)g(y), x,y C G, where G is a locally compact group, a is a continuous involution of G and # is an idempotent complex measure with compact support and which is a-invariant. We show that ∫Gg(xty)dp(t) + fcg(xta(y))dp(t) = 2g(x)g(y) if f = 0 and fcf(t.)dp(t) =0, where [fcf(t.)dp(t)](x) = fc f(tx)dμ(t). We also study some stability theorems of that equation and we establish the stability on noncommutative groups of the classical Wilson's functional equation f(xy) + X(y)f(xa(y)) = 2f(x)g(y) x, y C G, where X is a unitary character of G.展开更多
In this paper, among other things, the author studies the weak^*-closed left translation invariant complemented subspace of semigroup algebras and group algebras. Also, the author studies the relationships between pr...In this paper, among other things, the author studies the weak^*-closed left translation invariant complemented subspace of semigroup algebras and group algebras. Also, the author studies the relationships between projections and amenability.展开更多
In this paper, the concepts of approximate character amenability (contractibility), uniform approximate character amenability (contractibility) and w^*-approximate character amenability are introduced. We are con...In this paper, the concepts of approximate character amenability (contractibility), uniform approximate character amenability (contractibility) and w^*-approximate character amenability are introduced. We are concerned with the relations among the generalized concepts of character amenability for Banach algebras. We show that approximate character amenability, w^*-approximate character amenability and approximate character contractibility are the same properties, as uniform approximate character amenability and character amenability as uniform approximate character contractibility and character contractibility. The general theory for these concepts is also developed. Moreover, approximate character amenability of several concrete classes of Banach algebras related to locally compact groups and also some discrete semigroups is considered.展开更多
文摘For locally compact groups G, Kuan Yuan studied a notion of inner amenability groups, that is, if there exists an inner invariant mean on G. In this article, among other things, the author investigates the inner amenability on a locally compact group G. The author gives sufficient conditions and some necessary conditions about G to have an inner invariant mean.
文摘Let G be a locally compact group and let F be a closed subgroup of G × G. Pier introduced the notion of F-amenability which gives a new classification of groups. This concept generalizes the concept of amenability and inner amenability for locally compact groups. In this paper, among other things, we extend some standard results for amenable groups to F-amenable groups and give various characterizations for F-amenable groups. A sequence of characterizations of F-amenable groups is given here by developing the well-known Flner's conditions for amenable locally compact groups. Several characterizations of inner amenability are also given.
文摘In this note, we consider the multipliers on weighted function spaces over totally disconnected locally compact abelian groups (Vilenkin groups). Firstly we show an (H1 ,L ) multiplier result. We also give an (Hap ,Hap) multiplier result under a similiar condition of Lu Yang type. In section 2, we obtain a result about the boundedness of multipliers on weighted Besov spaces.
文摘Three classical compactification procedures are presented with nonstandard flavour. This is to illustrate the applicability of Nonstandard analytic tool to beginners interested in Nonstandard analytic methods. The general procedure is as follows: A suitable equivalence relation is defined on an enlargement <sup>*</sup><em>X </em>of the space <em>X</em> which is a completely regular space or a locally compact Hausdorff space or a locally compact Abelian group. Accordingly, every <em>f</em> in <em>C</em>(<em>X</em>,<em>R</em>) (the space of bounded continuous real valued functions on <em>X</em>) or <em>Cc</em>(<em>X</em>,<em>R</em>) (the space of continuous real valued functions on <em>X</em> with compact support) or the dual group <span style="white-space:nowrap;">Γ </span>of the locally compact Abelian group <em>G</em> is extended to the set <img alt="" src="Edit_b9535172-924d-44f0-bab3-c49db17a3b7a.png" /> of the above mentioned equivalence classes. A compact topology on <img alt="" src="Edit_9d7962a3-b8a3-4693-b62a-078c8c4b4853.png" /> is obtained as the weak topology generated by these extensions of <em>f</em>. Then <em>X</em> is naturally imbedded densely in <img alt="" src="Edit_f7d403b2-eff3-4555-b8e7-1b106e06d2e7.png" />.
基金This work was supported by the Key Program of the Natural Science Foundation of Fujian Province (No. 2020J02043)the National Natural Science Foundation of China (Grant No. 11571158)the Institute of Meteorological Big Data-Digital Fujian, and Fujian Key Laboratory of Data Science and Statistics.
文摘Two non-discrete Hausdorff group topologiesτandδon a group G are called transversal if the least upper boundτ⋁δofτandδis the discrete topology.In this paper,we discuss the existence of transversal group topologies on locally pseudocompact,locally precompact,or locally compact groups.We prove that each locally pseudocompact,connected topological group satisfies central subgroup paradigm,which gives an affirmative answer to a problem posed by Dikranjan,Tkachenko,and Yaschenko[Topology Appl.,2006,153:3338–3354].For a compact normal subgroup K of a locally compact totally disconnected group G,if G admits a transversal group topology,then G/K admits a transversal group topology,which gives a partial answer again to a problem posed by Dikranjan,Tkachenko,and Yaschenko in 2006.Moreover,we characterize some classes of locally compact groups that admit transversal group topologies.
文摘Let π be a unitary representation of a locally compact topological group G on a separable Hilbert space H. A vector ψ∈ H is called a continuous frame wavelet if there exist A, B 〉 0 such that A||Ф||2≤fG|〈π(G)ψ,Ф|2dg≤B||Ф||2(Ф∈H),in which dg is the left Haar measure of G. Similar to the study of wavelets, an essential problem in the study of continuous frame wavelets is how to characterize them under the given unitary representation. Moreover, we investigate a relation between admissible vectors of π and its components.
文摘In this paper we study the solutions and stability of the generalized Wilson's functional equation fc f(xty)dtt(t) + fc f(xtσ(y))dtt(t) =2f(x)g(y), x,y C G, where G is a locally compact group, a is a continuous involution of G and # is an idempotent complex measure with compact support and which is a-invariant. We show that ∫Gg(xty)dp(t) + fcg(xta(y))dp(t) = 2g(x)g(y) if f = 0 and fcf(t.)dp(t) =0, where [fcf(t.)dp(t)](x) = fc f(tx)dμ(t). We also study some stability theorems of that equation and we establish the stability on noncommutative groups of the classical Wilson's functional equation f(xy) + X(y)f(xa(y)) = 2f(x)g(y) x, y C G, where X is a unitary character of G.
文摘In this paper, among other things, the author studies the weak^*-closed left translation invariant complemented subspace of semigroup algebras and group algebras. Also, the author studies the relationships between projections and amenability.
基金supported by National Natural Science Foundation of China (Grant No. 11226125)
文摘In this paper, the concepts of approximate character amenability (contractibility), uniform approximate character amenability (contractibility) and w^*-approximate character amenability are introduced. We are concerned with the relations among the generalized concepts of character amenability for Banach algebras. We show that approximate character amenability, w^*-approximate character amenability and approximate character contractibility are the same properties, as uniform approximate character amenability and character amenability as uniform approximate character contractibility and character contractibility. The general theory for these concepts is also developed. Moreover, approximate character amenability of several concrete classes of Banach algebras related to locally compact groups and also some discrete semigroups is considered.