【作者】 尤超；

【导师】 游宏； V.M.Manuilov；

【作者基本信息】 哈尔滨工业大学 ， 基础数学， 2010， 博士

【摘要】 群的酉表示是代数学、几何学、泛函分析等数学分支的重要研究课题。近年来,群与C*-代数交叉的理论和应用的研究也日趋活跃。本文主要从群的酉表示入手,在（T）-性质群的几乎酉表示、自由群F₂的受限酉表示和相应的群C*-代数的刻画等方面展开了研究,同时也讨论了几乎收敛性在C*-代数和离散拓扑半群中的应用。所得的主要结果如下:1.将Z˙uk关于（T）性质的充分条件推广到了几乎酉表示的情形。这使得对（T）-性质群在几乎酉表示π下的平均算子π（χ）谱的研究不再依赖于渐进酉表示成为可能,从而将我们研究对象的范围拓展到非AGA的（T）-性质群。2.给出了自由群F₂的μ-受限酉表示的概念,指出了该类表示的大量存在性以及对参数μ的依赖性。利用μ-受限酉表示诱导出了对应的群C*-代数A_μ,并通过证明（A_μ,I,A/B）是一个C*-代数的连续丛,刻画了M_T?关于参数K的连续性。将A₀与一个C*-代数的融合积等同起来,应用Cuntz关于融合自由积的K-理论正合列计算出了A₀的K-群。再以K4和A₀的K-群相同作为基础,通过建立K2泛表示和平凡表示的同伦,借助于函子M_T?的同伦不变性计算出了M_T的K-群。3.分别从理论和应用两个方面对几乎收敛性展开了研究。理论方面,在N上定义了有限可加的概率测度M_TM_T,指出M_TM_T可测的序列,即恰当分布列都是几乎收敛的,而且其（唯一的）Banach极限可表示成一个形式积分。通过引入Banach极限泛函的概念,定义了赋范向量空间中有界序列的强几乎收敛性,指出现有文献中的几乎收敛和准几乎收敛与我们的强几乎收敛是等价的,从而对向量值序列几乎收敛的概念进行了总结和统一。应用方面,从几乎收敛列的空间M_T∞M_T（N,C）中提取出一个含幺交换C*-代数――M_T.通过Gelfand变换,证明M_T的极大理想空间KN是N的一个紧化,而且包含KN作为闭子集,M_T则为Hilbert C*-模理论提供了一个非C*-自反C*-代数的例子。最后还指出上述几乎收敛性的应用对于可数无限的左顺从可消半群同样适用。更多还原

【Abstract】 Unitary representation of group is an important research subject of algebra, geome-try, functional analysis and other branches of mathematics. Recently, research on interdis-ciplinary theory and application of group and C*-algebra has become more and more ac-tive. Starting with unitary representation of group, this dissertation studies almost unitaryrepresentation of Property （T） group, constrained unitary representation of free group F₂and corresponding group C*-algebras, applications of almost convergence to C*-algebraand discrete topological semi-group, etc. The main results obtained are as follows:1. Z˙uk’s sufficient condition for Property （T） is generalized to the case of almostunitary representation. This makes the study on the spectrum of averaging operatorπ（χ）of Property （T） group with almost unitary representationμpossibly independent fromasymptotic unitary representation, hence extends our research range to non-AGA Property（T） groups.2. The concept ofμ-constrained unitary representation is defined for free group F₂,and we show its existence, abundance and dependence on parameterμ. Correspondinggroup C*-algebraμμis induced byμ-constrained unitary representations. By provingthat （Aμ,I,A/B） is a continuous bundle of C*-algebras, it is shown that A_μs possesscertain continuity with respect toμ. We calculate theμ-groups of A0, by identifyingA₀ with some amalgamated free product of C*-algebras and applying Cuntz’sμ-theoryexact sequences about amalgamated free product. Based on the fact thatμ-groups ofA₀ and A₄ are the same, by constructing homotopy between universal representation andtrivial representation of F₂, we calculate the K-groups of Aμfrom homotopy invarianceof functor K_i.3. Both theory and application of almost convergence are studied respectively. Intheory, finitely additive probability measureμ_ac is defined on N, and it is pointed out thatμ_ac-measurable sequence, i.e., properly distributed sequence, is almost convergent withtheir （unique） Banach limit in the form of formal integral. By introducing the concept ofBanach limit functional, the concept of strong almost convergence is defined for boundedsequences in normed vector space, and it is shown that almost convergence and quasi-almost convergence in some papers are equivalent to our strong almost convergence here, hence we summarize and unify the theory of almost convergence of vector-valued se-quences. In application, we extract a unital commutative C*-algebra–M_T from the spacel^∞_ac（N,C） of almost convergent sequences. By Gelfand transform, it is proved that themaximal ideal space t_N of M_T is a compactification of N containingβN as a closedsubset, while M_T provides an example of non-C*-re?exive C*-algebra for Hilbert C*-module theory. Finally, we assert that applications of almost convergence above could begeneralized to countably infinite left amenable cancellative semi-groups.更多还原