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补充：
Operator algebra
（From Wikipedia, the free encyclopedia）
In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space with the multiplication given by the composition of mappings. Although it is usually classified as a branch of functional analysis, it has direct applications to representation theory, differential geometry, quantum statistical mechanics and quantum field theory.
Though algebras of operators are studied in various contexts (for example, algebras of pseudo-differential operators acting on spaces of distributions), the term operator algebra is usually used in reference to algebras of bounded operators on a Banach space or, even more specially in reference to algebras of operators on a separable Hilbert space, endowed with the operator norm topology.
In the case of operators on a Hilbert space, the adjoint map on operators gives a natural involution which provides an additional algebraic structure which can be imposed on the algebra. In this context, the best studied examples are self-adjoint operator algebras, meaning that they are closed under taking adjoints. These include C*-algebras and von Neumann algebras. C*-algebras can be easily characterized abstractly by a condition relating the norm, involution and multiplication. Such abstractly defined C*-algebras can be identified to a certain closed subalgebra of the algebra of the continuous linear operators on a suitable Hilbert space. A similar result holds for von Neumann algebras.
Commutative self-adjoint operator algebras can be regarded as the algebra of complex valued continuous functions on a locally compact space, or that of measurable functions on a standard measurable space. Thus, general operator algebras are often regarded as a noncommutative generalizations of these algebras, or the structure of the base space on which the functions are defined. This point of view is elaborated as the philosophy of noncommutative geometry, which tries to study various non-classical and/or pathological objects by noncommutative operator algebras.
这个课程一共有八讲，具体的安排如下：
第一讲 Hilbert 空间 (包括基本性质和其上投影)
第二讲 C*代数初步 （包括Gelfand表示， GNS构造等）
第三讲 C*代数例子 （UHF代数等）
第四讲 von Neumann 代数初步 （拓扑，双换位定理，Kaplansky定理）
第五讲 von Neumann 代数分类 （极分解定理，置换群因子，自由群因子）
第六讲 超有限II1型因子 （条件期望等）
第七讲 群测度空间构造 （具体因子构造）
第八讲 Jones 基本构造
参考读物
(1) Kadison and Ringrose, Fundamentals of the theory of operator algebras Vol. I and II, Academic Press, Inc. 1983 and 1986
(2) K.R. Davidson, C*-Algebras by Example. Fields Institute Monographs. 1991
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Post By：2011/9/17 23:59:24 [只看该作者]
