INVERTIBILITY PRESERVING LINEAR MAPS OF BANACH ALGEBRAS
INVERTIBILITY PRESERVING LINEAR MAPS OF BANACH ALGEBRAS
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保持Banach代数线性映射的不可逆性
DOI：
10.1090/conm/364/06677
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发表时间：
2001
期刊：
影响因子：
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通讯作者：
L. Harris
中科院分区：
文献类型：
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作者：
L. Harris
这个演讲讨论了R·V·卡迪森和我自己的一个猜想。我们的猜想是，一个单位C*-代数到另一个保持恒等式的一对一线性映射是Jordan同构的，如果它将第一个C*-代数的可逆元映射到另一个C*-代数的可逆元上。证明了这一猜想与Cartan唯一性定理之间的联系。1.定义和记法自始至终，A和B表示有单位元的复Banach代数(记为e)。Put Ainv={x∈A：x−1 Existes}，给定x∈A，设σ(X)={λ∈C：λe−x 6∈Ainv}表示x的谱，x的谱半径由|x|σ=sup{|λ|：λ∈σ(X)}定义。众所周知，σ(X)是非空紧集，并且|x|σ=Lim n→∞‖x‖≤‖x‖实际上，这些事实已经通过应用预解式的全纯性质得到了证明[12，p.125]。根据定义，如果对所有x∈A满足σ(Zx)={0}的唯一元素z∈A是z=0，则A是半单的。例如[12，Th.24.8.7]，如果A是Banach空间X上所有有界线性算子的Banach代数L(X)，则A是半单的。回想一下，C*-代数是某个希尔伯特空间H的L(H)的闭复子代数A，使得A包含它的每个元素的伴随。自始至终，首都1991年数学学科分类。初级46L05。作者希望对在以色列卡尔米尔举行的复杂分析和动力系统国际会议的组织者在会议期间的友好和热情款待表示感谢。
This talk discusses a conjecture of R. V. Kadison and myself. Our conjecture is that each one-to-one linear map of one unital C*-algebra onto another that preserves the identity is a Jordan isomorphism if it maps the invertible elements of the first C*-algebra onto the invertible elements of the other C*-algebra. Connections are shown between this conjecture and Cartan’s uniqueness theorem. 1. Definitions and notation Throughout, A and B denote complex Banach algebras with identity (denoted by e). Put Ainv = {x ∈ A : x−1 exists } and given x ∈ A, let σ(x) = {λ ∈ C : λ e− x 6∈ Ainv} denote the spectrum of x. The spectral radius of x is defined by |x|σ = sup{|λ| : λ ∈ σ(x)}. It is well known that σ(x) is a non-empty compact set and that |x|σ = lim n→∞ ‖x‖ ≤ ‖x‖ Indeed, these facts are proved by applying holomorphic properties of the resolvent [12, p. 125]. By definition, A is semisimple if the only element z ∈ A satisfying σ(zx) = {0} for all x ∈ A is z = 0. For example [12, Th. 24.8.7], if A is the Banach algebra L(X) of all bounded linear operators on a Banach space X, then A is semisimple. Recall that a C*-algebra is a closed complex subalgebra A of L(H) for some Hilbert space H such that A contains the adjoints of each of its elements. Throughout, capital 1991 Mathematics Subject Classification. Primary 46L05. The Author wishes to express his gratitude to the organizers of the International Conference on Complex Analysis and Dynamical Systems, Karmiel, Israel, for their kindness and hospitality during the conference.