共变完全多正线性映射的共变投射表示

研究了C*-代数中的共变完全多正线性映射,证明了共变完全多正线性映射可以诱导Hilbert C*-模上的共变投射表示,并且给出了共变完全多正线性映射的KS- GNS(Kasparov,Stinespring,Gel’fand,Naimark,Segal)构造.     

Pontrjagin空间上一般算子代数弱闭和一致闭的等价条件

本文研究Pontrjagin空间上一般算子代数弱闭和一致闭的等价条件,得到定理:设C0(U),C1(U,L,R,D,V),C2a(U),C2b(U,R),C3a(U),C3b(U,R)分别是Ⅱk空间上第0,Ⅰ,Ⅱa,Ⅱb,Ⅲa和Ⅲb类的算子代数,则(1)C0(U),C2a(U)或C3a(U)为一致闭(弱闭)的等价条件是U是Hibert空间G上的C*-代数(W*-代数;(2)C1(U,L,R,D,V)为一致闭(弱闭)的等价条件是U是Hibert空间H上的C*-代数(W*-代数),并且R是闭子空间,V是闭算子,L对称闭的;(3)C2b(U,R)或C3b(U,R)为一致闭(弱闭)的等价条件是U是Hibert空间H上的C*-代数(W*-代数),并且R是闭子空间．     

Simple generalized inductive limits of C*-algebras

We give a necessary and sufficient condition where a generalized inductive limit becomes a simple C~*-algebra. We also show that if a unital C~*-algebra can be approximately embedded into some tensorially self absorbing C~*-algebra C(e.g., uniformly hyperfinite(UHF)-algebras of infinite type, the Cuntz algebra O_2),then we can construct a simple separable unital generalized inductive limit. When C is simple and infinite(resp.properly infinite), the construction is also infinite(resp. properly infinite). When C is simple and approximately divisible, the construction is also approximately divisible. When C is a UHF-algebra and the connecting maps satisfy a trace condition, the construction has tracial rank zero.     

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.     

C~* -代数上完全正映射的刻画

本文给出C* -代数之间完全正映射的刻画,证明:如果A,B是有单位元的C*-代数,则映射Φ:A→B为完全正映射当且仅当存在保单位*-同态πA:A→B(K)、等距* -同态πB:B→B(H)及有界线性算子V:H→K,使得πB(Φ(1))=V*V 且■a∈A,都有πB(Φ(a))=V*π(a)V.作为推论,得到著名的Stinespring膨胀定理.     

WQ*-algebras of measurable operators

Every C*-algebra $\mathfrak{A}$ has a faithful *-representation π in a Hilbert space $\mathcal{H}$ . Consequently it is natural to pose the following question: under which conditions, the completion of a C*-algebra in a weaker than the given one topology, can be realized as a quasi *-algebra of operators? The present paper presents the possibility of extending the well known Gelfand — Naimark representation of C*-algebras to certain Banach C*-modules.     

Problem of invariant subspaces in C*-algebras

Let A be a separable simple C*-algebra. For each a(≠0) in A, there exists a separable faithful and irreducible * representation (π, Hπ) on A such that π(a) has a non-trivial invariant subspace in Hπ.     

On property of injectivity for real W*-algebras and JW-algebras

In this paper injective real W*-algebras are investigated. It is shown that injectivity is equivalent to the property of E (extension property). It is proven that a real W*-algebra is injective iff its hermitian part is injective, and it is equivalent to, that the enveloping W*-algebra is also injective. Moreover, it is shown that if the second dual space of a real C*-algebra is injective, then the real C*-algebra is nuclear.     

G-旋模型场代数中的对偶定理

设Ｇ是有限群，　Ｈ是Ｇ的子群，Ｄ（Ｇ）为Ｇ的Ｄｏｕｂｌｅ代数，　Ｆ是　Ｇ－旋模型所对应的场代数．　本文考虑Ｄ（Ｇ）的Ｈｏｐｆ子代数Ｄ（Ｈ），证明了Ｆ的Ｄ（Ｈ）不变子空间ＡＨ是Ｃ＊－代数．Ｄ（Ｈ）存在Ｃ＊－表示，使得Ｄ（Ｈ）和ＡＨ互为换位子．     

Reduced C*-algebras of Fell bundles over inverse semigroups

We construct a weak conditional expectation from the section C*-algebra of a Fell bundle over a unital inverse semigroup to its unit fibre. We use this to define the reduced C*-algebra of the Fell bundle. We study when the reduced C*-algebra for an inverse semigroup action on a groupoid by partial equivalences coincides with the reduced groupoid C*-algebra of the transformation groupoid, giving both positive results and counterexamples.     

Partial Maps with Domain and Range: Extending Schein's Representation

Let A be a *-algebra. An additive mapping E : A → A is called a Jordan *-derivation if E(X²) = E(x)x^*+xE(x) holds, for all x 6 A. These mappings have been extensively studied in the last 6 years by Bresar, Semrl, Vukman and Zalar because they are closely connected with the problem of representability of quadratic functionals by sesquilinear forms. This study was, however, always in the setting of associative rings. In the present paper we study Jordan *-derivations on the Cayley-Dickson algebra of octonions, which is not associative. Our first main result is that every Jordan *-derivation on the octonion algebra is of the form E(x)=ax*-xa. In the terminology of earlier papers this means that every Jordan *-derivation on the octonion algebra is inner. This generalizes the known fact that Jordan *-derivations on complex and quaternion algebras are inner. Our second main result is a representation theorem for quadratic functionals on octonion modules. Its proof uses the result mentioned above on Jordan *-derivations.     

非可换模糊逻辑系统PL~*及其完备性

首先建立了非可换R_0t-模,以此为语义背景将模糊逻辑形式系统L~*拓广到非可换情形,提出了新的模糊逻辑形式系统PL~*,证明了系统PL~*的可靠性定理．其次,引入PL~*-代数及其滤子概念,得到PL~*-代数的正规素滤子定理,借此证明了PL~*系统的完备性．最后说明了PR_0t-模及PL~*系统可能的应用方向．     

EXTENSION OF MULTIPLIERS AND INJECTIVE HILBERT MODULES

The author studies the problem whether a multiplier of a hereditary C^*-subalgebra B of a C^*-algebra A can be extended to a multiplier of A. One related problem is the Hahn-Banach extension theorem for Hilbert modules over C^*-algebras. It is shown that every self-dual Hilbert module over W^*-algebra or an injective C^*-algebra is injective.     

Induced Quadratic Modules

Positivity in *-algebras can be defined either algebraically, by quadratic modules, or analytically, by *-representations. By the induction procedure for *-representations, we can lift the analytical notion of positivity from a *-subalgebra to the entire *-algebra. The aim in this article is to define and study the induction procedure for quadratic modules. The main question is when a given quadratic module on the *-algebra is induced from its intersection with the *-subalgebra. This question is very hard even for the smallest quadratic module (i.e. the set of all sums of hermitian squares) and will be answered only in very special cases.     

非可换模糊逻辑系统PL~*及其完备性

首先建立了非可换R_0t-模,以此为语义背景将模糊逻辑形式系统L~*拓广到非可换情形,提出了新的模糊逻辑形式系统PL~*,证明了系统PL~*的可靠性定理．其次,引入PL~*-代数及其滤子概念,得到PL~*-代数的正规素滤子定理,借此证明了PL~*系统的完备性．最后说明了PR_0t-模及PL~*系统可能的应用方向．     

URS' For Weierstrass Products without Exponential Factors

Let $ \cal W $ be the set of entire functions equal to a Weierstrass product of the form $ {f(x)= Ax^q\lim_{r \to \infty} \prod_{|a_j|\leq r}{(1- \fraca {x} {a_j})}} $ where the convergence is uniform in all bounded subsets of $ {\shadC} $ , let $ \cal V $ be the set of $ f\in {\cal W} $ such that $ {\shadC} [\,f]\subset {\cal W} $ , and let $ {\cal H} $ be the $ {\shadC} $ -algebra of entire functions satisfying $ { {\lim_{r\to \infty } } ({\ln M(r,f) / r})=0} $ . Then $ \cal H $ is included in $ {\cal V} $ and strictly contains the set of entire functions of genus zero, (which, itself, strictly contains the $ {\shadC} $ -algebra of entire functions of order 𝜌 < 1). Let $ n, m\in {\shadN} ^* $ satisfy n > m S 3. Let $ a\in {\shadC}^* $ satisfies $ {a^n\not = \fraca{n^n}{(m^m(n-m)^{n-m}})} $ and assume that for every ( n m m )-th root ξ of 1 different from m 1, a satisfies further $ {a^{n}\neq (1+\xi )^{n-m} (\fraca{n^n}{((n-m)^{n-m}m^m}))} $ . Let P ( X ) = X n m aX m + 1 and let T n,m ( a ) be the set of its zeros. Then T n,m ( a ) has n distinct points and is a urs for $ {\cal V} $ . In particular this applies to functions such as sin x and cos x .     

可能行无限图C~*-代数

本文研究可能行无限有向图的C~*-代数。对于一个可能行无限的有向图E,通过引进集合S(μ,v),将行无限点上的算子拓扑强收敛关系代数化表示出来,并由此构造了一个结构丰富的非零*-代数H_E;进而利用H_E证明了一个由Cuntz-Krieger E-族{s_e,p_v}生成的泛C~*-代数 C~*(E)的存在性,并且证明了H_E和 C~*(E)在图同构意义下不依赖于E的选择,从而是可能行无限有向图的同构不变量。     

The automorphism group of the Toeplitz algebra on H2(T2)

The automorphism group of the C*-algebra generated by the Toeplitz operatos with symbols being continuous functions on the bicircle is characterized completely.The investigation is based on the analysis of the behaviour of an automorphism of theToeplitz algebra on its C*-ideal chains, and the state of the closed ideals in C(T)(×)(h)(H).     

The tiling C*-algebra viewed as a tight inverse semigroup algebra

We realize Kellendonk’s C*-algebra of an aperiodic tiling as the tight C*-algebra of the inverse semigroup associated to the tiling, thus providing further evidence that the tight C*-algebra is a good candidate to be the natural associative algebra to go along with an inverse semigroup.