A non-smooth extension of Frechet differentiability of the norm with applications to numerical ranges

Aparicio, C.; Ocaña, Francisco A.; Payá, Rafael; Rodríguez, Ángel Gutiérrez

doi:10.1017/s0017089500006443

Cited by 20 publications

(15 citation statements)

References 9 publications

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“…Actually, all slices (and, more generally, all nonempty relatively weakly open subsets) of the closed unit ball of L(H) have diameter equal to 2 (see [5] and [4]). The reader is referred to [14] for quantitative versions of the fact that the units of norm-unital normed algebras are strongly extreme points, and to [3,6,20,23,24] for other interesting geometrical properties of the units of norm-unital normed algebras.…”

Section: Proof Assertion (1) Is Straightforwardmentioning

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: Proof Assertion (1) Is Straightforwardmentioning

confidence: 99%

On multiplicatively closed subsets of normed algebras

Galindo

Palacios

2010

Journal of Algebra

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“…It is worth mentioning that the result just quoted need not remain true if the couple (A, 1) above is replaced with (X, e) for an arbitrary Banach space X and an element e in S X . Actually, if we denote by e the constant function equal to e on Γ , the couples (X, e) as above which satisfy V ( ∞ (Γ, X), e, φ) = co[ γ∈Γ V (X, e, φ(γ))] for every set Γ and every φ in ∞ (Γ, X) are characterized as those such that the norm of X is strongly subdifferentiable at e (see [1,Theorem 2.7] or [44,Corollary 2]). According to [28,Corollary 4.4], the strong subdifferentiability of a Banach space X at an element e in S X is also equivalent to the upper semicontinuity (n − n) of the duality mapping of X at e, previously introduced in [26].…”

Section: Characterizationsmentioning

confidence: 99%

Unitary Banach algebras

et al. 2004

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Abstract. In a Banach algebra an invertible element which has norm one and whose inverse has norm one is called unitary. The algebra is unitary if the closed convex hull of the unitary elements is the closed unit ball. The main examples are the C * -algebras and the 1 group algebra of a group. In this paper, different characterizations of unitary algebras are obtained in terms of numerical ranges, dentability and holomorphy. In the process some new characterizations of C * -algebras are given.

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“…In particular, if AT is a quite smooth dual Banach space whose duality mapping has weakly compact values, then X is reflexive-a result proved by Zhang [22,Theorem 2]. On the other hand, it was shown in [1] (see also [10,Theorem 3.3]) that a dual Banach space whose norm is strongly subdifferentiable on the unit sphere must be reflexive. This result has been improved by Godefroy [13], who shows that the assumption of being a dual space can be weakened by only assuming that it satisfies the so-called finite-infinite intersection property ( IP/,oo f°r short).…”

Section: Theorem Every Quite Smooth Banach Space Is An Asplund Spacementioning

confidence: 99%

“…The obvious relations between the above assertions are: (1) (1) iy iy (2) =P (3) <= (4) 4-v-4 (5) =► (6) <= (1) We intend to give examples showing that no other implication in the diagram is true. It is clearly enough to show that (2) ^ (7), (4) ^ (5), (5) £■ (3), and (7)^(3).…”

Section: N-»oomentioning

confidence: 99%