AN ASYMPTOTIC DOUBLE COMMUTANT THEOREM FOR C*-ALGEBRAS
AN ASYMPTOTIC DOUBLE COMMUTANT THEOREM FOR C*-ALGEBRAS
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DOI：
10.1090/s0002-9947-1978-0506620-0
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发表时间：
1978-10
影响因子：
1.3
通讯作者：
D. Hadwin
中科院分区：
文献类型：
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作者：
D. Hadwin
证明了冯诺依曼双交换定理的渐近版本，其中 C* 代数扮演冯诺依曼代数的角色。该定理用于研究相似性、自反性和还原性的渐近版本。结果表明，每一个不可分的、范闭的、可交换的、强还原的代数都是自伴的。应用到与正常（次正规）算子相似的算子的研究。特别是，如果 T 类似于正规（次正规）算子并且 «■ 是由 I 生成的 C* 代数的表示，则 ir(T) 类似于正规（次正规）算子。 1. 简介。冯·诺依曼代数理论成功的原因之一是 J. 冯·诺依曼的双重交换定理 (46)，它给出了算子自伴代数的弱闭包的另一种描述。本文的目的是证明双交换定理的渐近版本，该定理给出算子自共代数范数闭包的另一种描述。这个渐近双交换定理有助于统一各种算子理论概念（例如相似性、自反性、还原性）的渐近版本。应用到与正常（或次正常）算子相似的算子的研究。还证明了算子的强还原、不可分、可交换、范数闭代数是自共轭的。自始至终，H 表示可分离的无限维复休伯特空间，B(H) 表示 H 上的算子集（有界线性变换），%(H) 表示 H 上的紧算子集。同时 § 表示 B(H) 的可分离的非空子集。然而，在第 8 节中，H 和 S 的可分离性假设将被放弃。如果 § Q B(H)，则 §* = {S*: S G S}，#"(§>) 是由 1 和 § 生成的范数闭代数，&W(S) 是由 1 和 S 生成的弱闭代数，C*(§) 是由 1 和 S 生成的 C*-代数，W*(S ) 是由 1 和 § 生成的冯诺依曼代数
An asymptotic version of von Neumann's double commutant theorem is proved in which C*-algebras play the role of von Neumann algebras. This theorem is used to investigate asymptotic versions of simi- larity, reflexivity, and reductivity. It is shown that every nonseparable, norm closed, commutative, strongly reductive algebra is selfadjoint. Applications are made to the study of operators that are similar to normal (subnormal) operators. In particular, if T is similar to a normal (subnormal) operator and «■ is a representation of the C*-algebra generated by I, then ir(T) is similar to a normal (subnormal) operator. 1. Introduction. One of the reasons for the success of the theory of von Neumann algebras is J. von Neumann's double commutant theorem (46), which gives an alternate description of the weak closure of a selfadjoint algebra of operators. It is the purpose of this paper to prove an asymptotic version of the double commutant theorem that gives an alternate description of the norm closure of a selfadjoint algebra of operators. This asymptotic double commutant theorem helps to unify asymptotic versions of various operator-theoretic concepts (e.g., similarity, reflexivity, reductivity). Applications are made to the study of operators that are similar to normal (or subnormal) operators. Also a proof is given that a strongly reductive, nonseparable, commutative, norm closed algebra of operators is selfadjoint. Throughout, H denotes a separable, infinite-dimensional complex Hubert space, B(H) denotes the set of operators (bounded linear transformations) on H, and %(H) denotes the set of compact operators on H. Also § denotes a separable, nonempty subset of B(H). However, in §8 the separability assumptions on H and S will be dropped. If § Q B(H), then §* = {S*: S G S}, #"(§>) is the norm closed algebra generated by 1 and §, &W(S) is the weakly closed algebra generated by 1 and S, C*(§) is the C*-algebra generated by 1 and S, and W*(S ) is the von Neumann algebra generated by