Centralizers in semisimple algebras, and descent spectrum in Banach algebras

Haily, Abdelfattah; Kaidi, A.; Palacios, Ángel Rodríguez

doi:10.1016/j.jalgebra.2011.06.021

Cited by 6 publications

(2 citation statements)

References 11 publications

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“…Denote by K[a] the subalgebra of A generated by {1, a}. Since a is algebraic, there exists an idempotent p ∈ K[a] such that a(1 − p) is nilpotent, and ap is invertible in K[a]p (see [12,Lemma 3.2]). Now it is enough to show that p = 0.…”

Section: Proposition 58 Let a Be A Nilpotent Normed Algebra Then Amentioning

confidence: 99%

On multiplicatively closed subsets of normed algebras

Galindo

Palacios

2010

Journal of Algebra

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It is well known that, if S is a bounded and multiplicatively closed subset of an associative normed algebra (A, · ), then there exists an equivalent algebra norm ||| · ||| on A such that |||s||| 1 for every s ∈ S. Although associativity is not an essential requirement in this result, it is easy to find examples of nonassociative normed algebras A where such a result fails. Actually, it can fail even if the subset S is reduced to a nonzero idempotent. We prove that it remain true in the nonassociative setting whenever the subset S is assumed to be contained in the nucleus of A. In the particular case that the subset S reduces to a nonzero nuclear idempotent p, we show that the equivalent algebra norm ||| · ||| above can be chosen so that p becomes a strongly exposed point of the closed unit ball of (A, ||| · |||). We study those (possibly nonassociative) normed algebras A satisfying the "norm-one boundedness property" (in short, NBP), which means that, as happened in the associative case, for every bounded and multiplicatively closed subset S of A, there exists an equivalent algebra norm ||| · ||| on A such that |||s||| 1 for every s ∈ S. We show that absolute-valued algebras, JB-algebras, and nilpotent normed algebras fulfil the NBP. We also show that, if an anti-commutative complete normed algebraic algebra A satisfies the NBP, then there exists n ∈ N such that L n a = 0 for every a ∈ A,where L a denotes the operator of left multiplication by a. It follows from a celebrated theorem of E.I. Zel'manov on the so-called Engel Lie algebras that a complete normed algebraic Lie algebra satisfies the NBP if and only if it is nilpotent.

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Section: Proposition 58 Let a Be A Nilpotent Normed Algebra Then Amentioning

confidence: 99%

On multiplicatively closed subsets of normed algebras

Galindo

Palacios

2010

Journal of Algebra

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“…Indeed, if X is actually a complex Banach space, and if the linear operator T above is bounded, then some power of T has finite rank if (and only if) the equality σ desc (T + S) = σ desc (S) holds for every bounded linear operator S on X commuting with T . Very recently, we have proved in [10] a variant of the fact just reviewed, where an arbitrary semisimple complex Banach algebra A replaces the algebra BL(X) of all bounded linear operators on the complex Banach space X, the socle of…”

Section: Introductionmentioning

confidence: 97%

Algebra descent spectrum of operators

2010

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We say that a Banach space X satisfies the "descent spectrum equality" (in short, DSE) whenever, for every bounded linear operator T on X, the descent spectrum of T as an operator coincides with the descent spectrum of T as an element of the algebra of all bounded linear operators on X. We prove that the DSE is fulfilled by 1 , all Hilbert spaces, and all Banach spaces which are not isomorphic to any of their proper quotients (so, in particular, by the hereditarily indecomposable Banach spaces [8]), but not by p, for 1 < p ≤ ∞ with p = 2. Actually, a Banach space is not isomorphic to any of its proper quotients if and only if it is not isomorphic to any of its proper complemented subspaces and satisfies the DSE.

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Topologically nilpotent normed algebras

Galindo

Palacios

2012

Journal of Algebra

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Centralizers in semisimple algebras, and descent spectrum in Banach algebras

Cited by 6 publications

References 11 publications

On multiplicatively closed subsets of normed algebras

On multiplicatively closed subsets of normed algebras

Algebra descent spectrum of operators

Topologically nilpotent normed algebras

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