Displaying similar documents to “A remark on C * -algebras”

Additively spectral-radius preserving surjections between unital semisimple commutative Banach algebras

Osamu Hatori, Go Hirasawa, Takeshi Miura (2010)

Open Mathematics

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Let A and B be unital, semisimple commutative Banach algebras with the maximal ideal spaces M A and M B, respectively, and let r(a) be the spectral radius of a. We show that if T: A → B is a surjective mapping, not assumed to be linear, satisfying r(T(a) + T(b)) = r(a + b) for all a; b ∈ A, then there exist a homeomorphism φ: M B → M A and a closed and open subset K of M B such that T a ^ y = T e ^ y a ^ φ y y ∈ K T e ^ y a ^ φ y ¯ y ∈ M ℬ ∖ K for all a ∈ A, where e is unit element of A. If, in addition, T e ^ = 1 and T i e ^ = i on M B, then T is an algebra isomorphism. ...

Normed "upper interval" algebras without nontrivial closed subalgebras

C. J. Read (2005)

Studia Mathematica

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It is a long standing open problem whether there is any infinite-dimensional commutative Banach algebra without nontrivial closed ideals. This is in some sense the Banach algebraists' counterpart to the invariant subspace problem for Banach spaces. We do not here solve this famous problem, but solve a related problem, that of finding (necessarily commutative) infinite-dimensional normed algebras which do not even have nontrivial closed subalgebras. Our examples are incomplete normed...