Continuity of Lie Isomorphisms of Banach Algebras

We prove that if A and B are semisimple Banach algebras, thenthe separating subspace of every Lie isomorphism from A ontoB is contained in the centre of B. 1991 Mathematics SubjectClassification 17B40, 17B60, 46H40.     

一类单完备李代数

确定了代数Va,b.c的导子代数,其中Va.b.c是由非交换代数Ca,b,c[X±1,Y±1,Z±1]的所有内导子所张成的李代数,证明了导子代数数Va.b,c是由所有内导子和3个外导子D1,0,0,DO,1,0,D0,0,1张成的,同时还给出了一类无限维的单完备李代数.     

一类无穷矩阵代数的乘法序列连续性

设λ，μ是两个序列空间并有符号弱滑脊性，（λ，μ）是变换λ进入μ的无穷矩阵算子所成的无穷矩阵代数，本文研究了这类代数的强，Mackey、弱乘法序列连续性问题。     

Continuity of Derivations, Intertwining Maps, and Cocycles from Banach Algebras

Let A be a Banach algebra, and let E be a Banach A-bimodule.A linear map S:AE is intertwining if the bilinear map is continuous, and a linear map D:AE is a derivation if ¹D=0,so that a derivation is an intertwining map. Derivations fromA to E are not necessarily continuous. The purpose of the present paper is to prove that the continuityof all intertwining maps from a Banach algebra A into each BanachA-bimodule follows from the fact that all derivations from Ainto each such bimodule are continuous; this resolves a questionleft open in [1, p. 36]. Indeed, we prove a somewhat strongerresult involving left- (or right-) intertwining maps.     

Semisimplicity of Some Class of Operator Algebras on Banach Space

Let G be a locally compact abelian group and let be a representation of G by means of isometries on a Banach space. We define W_T as the closure with respect to the weak operator topology of the set where is the Fourier transform of f ∈L¹(G) with respect to the group T. Then W_T is a commutative Banach algebra. In this paper we study semisimlicity problem for such algebras. The main result is that if the Arveson spectrum sp(T) of T is scattered (i.e. it does not contain a nonempty perfect subset) then the algebra W_T is semisimple. Some related problems are also discussed.     

Isometric Isomorphisms Between Banach Algebras Related to a Certain Class of Clifford Topological Semigroups

In the present paper it is shown that if S₁ and S₂ are two Clifford topological semigroups satisfying certain conditions and T is an isometric isomorphism of LUC(S₁*) onto LUC(S₂*), then T maps S₁ topologically isomorphically onto S₂. Furthermore, T maps M ^l _n(S₁) (M(S₁), respectively) isometrically isomorphically onto M ^l _n(S₂) (M(S₂), respectively). Indeed, we have obtained a generalization of a well-known result of Ghahramani, Lau and Losert for locally compact groups to a more general setting of Clifford topological semigroups.     

New Results on Generalized $c$-Distance Without Continuity in Cone $b$-Metric Spaces over Banach Algebras

In this work, some new fixed point results for generalized Lipschitz mappings on generalized $c$-distance in cone $b$-metric spaces over Banach algebras are obtained, not acquiring the condition that the underlying cone should be normal or the mappings should be continuous. Furthermore, the existence and the uniqueness of the fixed point are proven for such mappings. These results greatly improve and generalize several well-known comparable results in the literature. Moreover, some examples and an application are given to support our new results.     

On a Spectral Condition in Banach Algebras

We use a spectral condition to characterise ideals and scalarsin a Banach algebra, and to study the additive properties ofthese ideals.     

论顺从Banach代数

In this paper we extend the concept of amenable algebra given by Johnson in [ 3 ] to n-amenable algebra. We prove several properties which are extension of 1-amenable algebras appeared in [3] and [4], and we give a sufficient condition for a Banach algebra being 2- amenable.     

Ideal Spaces of Banach Algebras

The ideal space Id(A) of a Banach algebra A is studied as abitopological space Id(A), _u, _n, where _u is the weakest topologyfor which all the norm functions I || a + I|| (with a A andI Id(A)) are upper semi-continuous, and _n is the de Groot dualof _u. When A is separable, _nu is either a compact, metrizabletopology, or it is neither Hausdorff nor first countable. TAF-algebrasare shown to exhibit the first type of behaviour. Applicationsto Banach bundles (which motivate the study), and to PI-Banachalgebras, are given. 1991 Mathematics Subject Classification:46H10, 46J20.     

K-Theory for Algebras of Operators on Banach Spaces

It is proved that, for each pair (m, n) of non-negative integers,there is a Banach space X for which K₀(B(X))Z^m and K₁(B(X))Zⁿ.The K-groups of all closed ideals of operators contained inthe ideal of strictly singular operators are computed, and someresults about the existence of splittings of certain short exactsequences are derived.     

Regularities in Noncommutative Banach Algebras

In this paper we introduce regularities and subspectra in a unital noncommutative Banach algebra and prove that there is a correspondence between them similar to the commutative case. This correspondence involves a radical on a class of Banach algebras equipped with a subspectrum. Taylor and Slodkowski spectra for noncommutative tuples of bounded linear operators are the main examples of subspectra in the noncommutative case.      

Differential Equations in Banach Algebras

Doklady Mathematics - In a complex Banach algebra that is not assumed to be commutative, nth-order linear differential equations with constant coefficients are considered. The corresponding...     

Cocycle Perturbation on Banach Algebras

Let $α$ be a flow on a Banach algebra$\mathcal{B}$, and $t → u_t$ a continuous function from$\boldsymbol{R}$into the group of invertible elements of$\mathcal{B}$such that $u_sα_s(u_t) = u_{s+t}, s, t ∈ \boldsymbol{R}$. Then $β_t$ = Ad$u_t ◦ α_t$, $t ∈ \boldsymbol{R}$ is also a flow on $\mathcal{B}$, where Ad$u_t(B) \triangleq u_tBu^{−1}_t$ for any $B ∈ \mathcal{B}$. $β$ is said to be a cocycle perturbation of $α$. We show that if $α$, $β$ are two flows on a nest algebra (or quasi-triangular algebra), then $β$ is a cocycle perturbation of $α$. And the flows on a nest algebra (or quasi-triangular algebra) are all uniformly continuous.     

Spectrum-Preserving Linear Mappings between Banach Algebras or Jordan-Banach Algebras

Spectrum-preserving linear mappings were studied for the firsttime by G. Frobenius [18]. He proved that a linear mapping     

Ideal Connes-Amenability of Dual Banach Algebras

Following Runde, we define the concept of ideal Connes-amenability for dual Banach algebras. For an Arens regular dual Banach algebra \({\mathcal {A}}\), we prove that the ideal Connes-amenability of \(\mathcal {A^{**}}\), the second dual of \({\mathcal {A}}\) necessities ideal Connes-amenability of \({{\mathcal {A}}}\). As a typical example, we show that von Neumann algebras are always ideally Connes-amenable. For a locally compact group G, the Fourier–Stieltjes algebra of G is ideally Connes-amenable, but not ideally amenable.