Invertibility Preserving Linear Maps of Banach Algebras
نویسنده
چکیده
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.
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