The uniqueness of the (complete) norm topology

Johnson, Brenda

doi:10.1090/s0002-9904-1967-11735-x

Cited by 84 publications

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“…P is a primitive ideal of A, and therefore P is closed in A. A/P is a Banach algebra with norm \\a + P||' A , aeA. Given αei, define S α+P (b+Λf) = αδ + j|f, δ e M. Then α + P -> 5 α+P is a faithful strictly irreducible representation of A/P into the bounded operators on B -M. Then a theorem of B. E. Johnson [6,Theorem 1,p. 537] implies that a + p-*Sa+r is a continuous map.…”

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confidence: 99%

Banach algebras which are ideals in a Banach algebra

Barnes¹

1971

Pacific J. Math.

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confidence: 99%

Banach algebras which are ideals in a Banach algebra

Barnes¹

1971

Pacific J. Math.

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“…Since A is prime, we conclude that a = 0. From Johnson's uniqueness-of-norm theorem [5] we deduce that the involution of A is continuous. Accordingly, K A is closed in A.…”

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Continuity of Lie mappings of the skew elements of Banach algebras with involution

Berenguer

Villena

1998

Proc. Amer. Math. Soc.

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Abstract. Let A and B be centrally closed prime complex Banach algebras with linear involution. If A is semisimple, then any Lie derivation of the skew elements of A is continuous and any Lie isomorphism from the skew elements of B onto the skew elements of A is continuous.The Lie product [a, b] = ab − ba induces on any Banach algebra A a Lie structure of great interest for their intimate connections with the geometry of manifolds modeled on Banach spaces. In case A has a linear involution * , then the skew elements are the linear subspace K A = {a ∈ A : a * = −a} which is a Lie subalgebraExamples 1. Let H be a complex Hilbert space. Let L(H) denote the primitive C * -algebra of all continuous linear operators on H, and for each a ∈ L(H), let a• denote the usual adjoint operator of a. 1. If J is a conjugation of H, then it is easy to check that the mapping * from L(H) to itself defined by a * = Ja • J is a linear involution on L(H). If J is an anticonjugation of H, then the mapping a * = −Ja • J is a linear involution on L(H). The skew elements relative to the preceding involutions are classical complex Banach-Lie algebras of bounded operators (see [3]).2. Let us denote by C ∞ the set of all compact linear operators on H and let · ∞ be the usual operator norm. For 1 ≤ p < ∞, let C p denote the usual class of those compact linear operators a on H for which a p = (where {µ n } is the sequence of eigenvalues of the operator (a • a) 1/2 arranged in decreasing order and repeated according to multiplicity. According to [2, Lemmas XI.9, XI.10, and XI.14], C p is a two-sided ideal of L(H) which becomes a complex Banach algebra for the norm · p . Since C p contains all the continuous linear operators with finitedimensional range, we deduce that C p is primitive. The involutions introduced in the preceding example leave invariant C p , and their skew elements are classical complex Banach-Lie algebras of compact operators (see [3]).It was proved in [3] that Lie derivations and Lie •-automorphisms of all the preceding Banach-Lie algebras are continuous.

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“…We sketch the proof and note its similarity to the proof of [4,Theorem 2 ]. Let 6 be a Jordan homomorphism from a Banach algebra A onto a semisimple Banach algebra B.…”

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“…Therefore we may assume that wd is a homomorphism of a Banach algebra onto the primitive algebra ir(B), and is thus continuous from A into the Banach algebra of bounded linear operators on the representation space of 7r by [4,Theorem l]. This implies that ir(y)=0.…”

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