Universal C*-algebras defined by completely bounded unital homomorphisms

Suppose A is a unital C*-algebra and r 1.In this paper,we define a unital C*-algebra C_(cb)*(A,r) and a completely bounded unital homomorphism α_r:A → C_(cb)*(A,r)with the property that C_(cb)*(A,r)=C*(α_r(A))and,for every unital C*-algebra B and every unital completely bounded homomorphism φ:A→ B,there is a(unique)unital *-homomorphism π:C_(cb)*(A,r)→B such thatφ=πoα_r.We prove that,if A is generated by a normal set {t_λ:λ∈Λ},then C_(cb)*(A,r)is generated by the set {α_r(t_λ):λ∈Λ}.By proving an equation of the norms of elements in a dense subset of C_(cb)*(A,r)we obtain that,if Β is a unital C*-algebra that can be embedded into A,then C_(cb)*(B,r)can be naturally embedded into C(cb)*(A,r).We give characterizations of C_(cb)*(A,r)for some special situations and we conclude that C_(cb)*(A,r)will be "nice" when dim(A)≤ 2 and "quite complicated" when dim(A)≥ 3.We give a characterization of the relation between K-groups of A and K-groups of C_(cb)*(A,r).We also define and study some analogous of C_(cb)*(A,r).     

算子代数上的Jordan初等映射(英文)

给定两个环R,R’.对于满足一定条件的环R,本文证明了若M:R→R’,M*:R’→R为满射且对A,C∈R和B,D∈R’满足M（AM*（B）C+CM*（B）A）=M（A）BM（C）+M（C）BM（A）,M*（BM（A）D+DM（A）B）=M*（B）AM*（D）+M*（D）AM*（B）则M和M*是可加的;若R和R’分别包含单位I和I’,M（I）,M*（I’）可逆,则存在环同构N使得M（A）=N（A）M（I）,M*（B）=N^-1（BM（I））.特别地,若R=R’为标准算子代数或Hilbert空间套代数,则M和M*可加且存在有界可逆的线性或共轭线性算子S和T使得M（A）=SAT,M*（B）=TBS或M（A）=TA*S,M*（B）=（SBT）*对任意的A,B∈R成立.     

Smarandache函数的几类相关方程的解

设φ(n),S(n)分别表示正整数n的Euler函数和Smarandache函数,利用初等的方法和技巧,依据Smarandache函数计算公式,给出k的方程φ(p~αm)=S(p~(ακ))的所有解,其中p为素数,α,m为正整数且gcd(m,p)=1,由此得到方程φ(n)=S(n~k)的所有解(n,k)进而确定了满足条件S(n)|σ(n)的全部正整数n.最后,根据莫比乌斯变换反演定理证明了方程φ(n)=∑_(d|n)S(d)仅有两个解,分别为n=2~5和n=3×2~5.     

从双曲α-Bloch 空间到双曲{\boldmath QK型空间上的复合算子

若φ 为单位圆盘D上的解析自映射, X为D上解析函数全体构成的Banach空间.定义X上复合算子C_φ: C_φ (f)=fοφ, 对任意 f∈X. 该文研究了从双曲α-Bloch 空间到双曲Q_K型空间上复合算子的有界性的特征. 另外, 还给出了从D_p,α 到Q_K(p, q) 空间上复合算子的有界性和紧性的特征.     

广义Bergman空间上的紧复合算子

首先证明广义Bergman空间A_(N,α)~p,(α-n-1,p0)上的复合算子C_φ的有界性和紧性是不依赖于p的,进而证明了若对某个q0和-n-1βα,C_φ在A_(N,β)~α上有界,则C_φ在A_(N,α)~p,α(α-n-1,p0)上是紧的当且仅当lim|z|→1-1-(|z|~2/1-|φ(1)|~2)=0.     

Characterizations of Centralizable Mappings on Algebras of Locally Measurable Operators

A linear mapping φ from an algebra A into its bimodule M is called a centralizable mapping at G ∈ A if φ(AB)=φ(A)B=Aφ(B) for each A and B in A with AB=G. In this paper, we prove that if M is a von Neumann algebra without direct summands of type I₁ and type II, A is a *-subalgebra with M ⊆ A ⊆ LS(M) and G is a fixed element in A, then every continuous (with respect to the local measure topology t(M)) centralizable mapping at G from A into M is a centralizer.     

自伴标准算子代数上强保持k-斜交换性映射的刻画

令H是维数大于2的复Hilbert空间,A是H上自伴标准算子代数.对于给定的正整数k≥1,H上算子A与B的k-斜交换子递推地定义为_*[A,B]_k=_*[A,_*[A,B]_k-1],其中_*[A,B]₀=B,_*[A,B]₁=AB-BA^*.设k≥4,φ是A上的值域包含所有一秩投影的映射.本文证明了φ满足_*[φ（A）,φ（B）]_k=_*[A,B]_k对任意A,B∈A都成立的充分必要条件是φ（A）=A对任意A∈A都成立,或φ（A）=-A对任意A∈A都成立.当k是偶数时后一情形不出现.     

Hilbert C~*-模上可共轭算子的并联和

本文研究了Hilbert C~*-模上可共轭算子的并联和,推广了矩阵和Hilbert空间上有界线性算子的一些相关结果.通过举例说明:存在一个Hilbert C~*-模H,以及H上的两个可共轭的正算子A和B,使得算子方程A~(1/2)=(A+B)~(1/2)X, X∈■(H)无解,其中■(H)为H上的可共轭算子全体.     

Additive Maps Preserving Nilpotent Perturbation of Scalars

Let X be a Banach space over F(= R or C) with dimension greater than 2. Let N(X) be the set of all nilpotent operators and B_0(X) the set spanned by N(X). We give a structure result to the additive maps on FI + B_0(X) that preserve rank-1 perturbation of scalars in both directions. Based on it, a characterization of surjective additive maps on FI + B_0(X) that preserve nilpotent perturbation of scalars in both directions are obtained. Such a map Φ has the form either Φ(T) = cAT A~(-1)+ φ(T)I for all T ∈ FI + B_0(X) or Φ(T) = cAT*A~(-1)+ φ(T)I for all T ∈ FI + B_0(X), where c is a nonzero scalar,A is a τ-linear bijective transformation for some automorphism τ of F and φ is an additive functional.In addition, if dim X = ∞, then A is in fact a linear or conjugate linear invertible bounded operator.     

N_(ψ,φ)-type Quotient Modules over the Bidisk

Let H~2■ be the Hardy space over ~the bidisk ■, and let M_(ψ,φ)=[(ψ(z)-φ(w))~2] be the submodule generated by(ψ(z)-φ(w))~2, where ψ(z) and φ(w) are nonconstant inner functions.The related quotient module is denoted by ■. In this paper, we give a complete characterization for the essential normality of N_(ψ,φ). In particular, if ψ(z) = z, we simply write M_(ψ,φ)and N_(ψ,φ) as M_φ and N_φ respectively. This paper also studies compactness of evaluation operators L(0)|N_φand R(0)|N_φ, essential spectrum of compression operator S_z on N_φ, essential normality of compression operators S_z and S_w on N_φ.     

多维非退化扩散过程的象集与图集的一致Hausdorff维数

设X(t)=X(0)+∫^t_0α(X(s))dB(s)+∫^t_0β( X(s))ds为一d(d≥3)维非退化扩散过程。令X(E)={X(t): t∈E}, GRX(E)={(t,X(t)): t∈E}，该文证明了：对几乎所有ω：E B([0,∞)),有dimX(E,ω)=dimGRX(E,ω)=2dimE,这里dimF表示F的Hausdorff维数。     

保持算子乘积投影的线性映射

设B(H)是复Hilbert空间H上的有界线性算子全体且dim H≥2.本文证明了B(H)上的线性满射φ保持两个算子乘积非零投影性的充分必要条件是存在B(H)中的酉算子U以及复常数λ满足λ~2=1,使得φ(X)=λU~*XU,(?)X∈B(H).同时也得到了线性映射保持两个算子Jordan三乘积非零投影的充分必要条件.     

剩余有限minimax可解群的自同构

设G是剩余有限minimax可解群,α是G的自同构且φ:G→G(g→[g,α])是满射,则有以下结果:(1)当α~p=1时,G是幂零类不超过h(p)的幂零群的有限扩张,其中h(p)是只与p有关的函数;(2)当α~4=1时,G存在一个指数有限的特征子群H,使得H″≤Z(H)和C_H(α~2)是Abel群.并且C_G(α~2)和G/[G,α~2]都是Abel群的有限扩张.     

振荡超奇性Hilbert变换的Sobolev有界性

主要研究R~n上沿曲线Γ(t)=(t~(p_1),t~(p_2),…,t~(p_n))的振荡超奇性Hilbert变换H_(n,α,β)=∫_0~1 f(x-Γ(t))e~(it-β)t~(-1-α),在Sobolev空间上的有界性,其中0p_1P_2…P_n,αβ0.证明了对于0γ(nα)/((n+1))(p_1+α),当|1/p-1/2|(β-(n+1)[α-(β+p_1)γ])/(2β)时,H_(n,α,β)是从L_γ~2(R~n))到L~2(R~n)的有界算子.特别地,当β≥(α-γp_1)/(γ+1/(n+1))等时,H_(n,α,β)是从L_γ~2(R~n)到L~2(R~n)的有界算子·     

反对角算子矩阵及其平方的(ω)性质的摄动

设H为复的无限维可分的Hilbert空间,B(H)为H上的有界线性算子的全体.若σ_a(T)\σ_(ea)(T)=π_(00)(T),则称T∈B(H)满足(ω)性质,其中σ_a(T)和σ_(ea)(T)分别表示算子T的逼近点谱和本质逼近点谱,π_(00)(T)={λ∈isoσ(T):0dimN(T-λI)∞}.T∈B(H)称为满足(ω)性质的摄动,若对任意的紧算子K,T+K满足(ω)性质.本文证明了反对角算子矩阵及其平方具有(ω)性质的摄动的等价性.     

A Bishop–Phelps–Bollobás Theorem for Asplund Operators

By characterizing Asplund operators through Fréchet differentiability property of convex functions, we show the following Bishop–Phelps–Bollobás theorem: Suppose that X is a Banach space,T : X → C(K) is an Asplund operator with ║T║= 1, and that x_0 ∈ S_X, 0 ε satisfy ║T(x_0)║ 1-ε~2/2.Then there exist x_ε∈ S_X and an Asplund operator S : X → C(K) of norm one so that ║S(x_ε)║ = 1, x_0-x_ε ε and ║T-S║ ε.Making use of this theorem, we further show a dual version of Bishop–Phelps–Bollobás property for a strong Radon–Nikodym operator T : ?_1 → Y of norm one: Suppose that y_0~*∈ S_(Y~*), ε≥ 0 satisfy T~*(y_0~*) 1-ε~2/2. Then there exist y_ε~*∈ S_(Y~*), x_ε∈(±e_n), y_ε∈ S_Y, and a strong Radon–Nikodym operator S : ?_1 → Y of norm one so that (ⅰ)║S(x_ε)║= 1;(ⅱ) S(x_ε) = y_ε;(ⅲ)║T-S║ ε;(ⅳ)║S~*(y_ε~*)║=y_ε~*, y_ε= 1;(ⅴ)║y_0~*-y_ε~*║ ε and (ⅵ)║T~*-S~*║ ε,where(e_n) denotes the standard unit vector basis of ?_1.     

K-POTENT PRESERVING LINEAR MAPS

Let B(X) be the Banach algebra of all bounded linear operators on a complex Banach space X. Let k ≥ 2 be an integer and φ a weakly continuous linear surjective map from B(X) into itself. It is shown that φ is k-potent preserving if and only if it is k-th-power preserving, and in turn, if and only if it is either an automorphism or an antiautomorphism on B(X) multiplied by a complex number λ satisfying λk-1= 1. Let A be a von Neumann algebra and B be a Banach algebra, it is also shown that a bounded surjective linear map from A onto B is k-potent preserving if and only if it is a Jordan homomorphism multiplied by an invertible element with (k - l)-th power I.     

三阶复正交矩阵与二阶矩阵代数上的相似变换

设M_2是二阶复矩阵的全体,Φ是M_2上的线性映射.本文建立了三阶复正交矩阵与M_2上的相似变换之间的一一对应关系,并利用这一对应关系证明了Φ保Lie积行列式(谱、边缘谱)的充要条件是存在c∈{±1,±i},二阶可逆矩阵T和二阶矩阵S,使得Φ(A)=cTAT~(-1)+tr(SA)I对所有A∈M_2都成立.     

多项式零点保持线性映射

设H是维数大于2的复Hilbert空间,β(H)代表H上所有有界线性算子全体．假定Φ是从β(H)到其自身的弱连续线性双射．我们证明了映射Φ满足对所有的A,B∈β(H),AB=BA~*蕴涵Φ(A)Φ(B)=Φ(B)Φ(A)~*当且仅当存在非零实数c和酉算子U∈(?)(H),使得Φ(A)=cUAU~*对所有的A∈β(H)成立．     

von Neumann代数上的局部可乘Lie n-导子

A₁，…，A_n的（n-1）-换位子记为p_n（A₁，…，A_n）.令M是von Neumann代数，n ≥ 2是任意正整数，L：M → M是一个映射.本文证明了，若M不含I₁型中心直和项，且L满足L（p_n（A₁，…，A_n））=∑_k=1ⁿ p_n（A₁，…，A_k-1，L（A_k），A_k+1，…，A_n）对所有满足条件A₁A₂=0的A₁，A₂，…，A_n ∈ M成立，则L（A）=φ（A）+f（A）对所有A ∈ M成立，其中φ：M → M和f：M → Z（M）（M的中心）是两个映射，且满足φ在P_iMP_j上是可加导子，f（p_n（A₁，A₂，…，A_n））=0对所有满足A₁A₂=0的A₁，A₂，…，A_n ∈ P_iMP_j成立（1 ≤ i，j ≤ 2），P₁ ∈ M是core-free投影，P₂=I-P₁；若M还是因子且n ≥ 3，则L满足条件L（p_n（A₁，A₂，…，A_n））=∑_k=1ⁿ p_n（A₁，…，A_k-1，L（A_k），A_k+1，…，A_n）对所有满足A₁A₂A₁=0的A₁，A₂，…，A_n ∈ M成立当且仅当L（A）=φ（A）+h（A）I对所有A ∈ M成立，其中φ是M上的可加导子，h是M上的泛函且满足h（p_n（A₁，A₂，…，A_n））=0对所有满足条件A₁A₂A₁=0的A₁，A₂，…，A_n ∈ M成立.