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Unitary Banach algebras

Julio Becerra Guerrero, Simon Cowell, Ángel Rodríguez Palacios, Geoffrey V. Wood (2004)

Studia Mathematica

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 ℓ₁ 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.

Universal spaces for strictly convex Banach Spaces.

Gilles Godefroy (2006)

RACSAM

We show that if a separable Banach space X contains an isometric copy of every strictly convex separable Banach space, then X contains an isometric copy of l1 equipped with its natural norm. In particular, the class of strictly convex separable Banach spaces has no universal element. This provides a negative answer to a question asked by J. Lindenstrauss.

Universal stability of Banach spaces for ε -isometries

Lixin Cheng, Duanxu Dai, Yunbai Dong, Yu Zhou (2014)

Studia Mathematica

Let X, Y be real Banach spaces and ε > 0. A standard ε-isometry f: X → Y is said to be (α,γ)-stable (with respect to T : L ( f ) ≡ s p a n ¯ f ( X ) → X for some α,γ > 0) if T is a linear operator with ||T|| ≤ α such that Tf- Id is uniformly bounded by γε on X. The pair (X,Y) is said to be stable if every standard ε-isometry f: X → Y is (α,γ)-stable for some α,γ > 0. The space X[Y] is said to be universally left [right]-stable if (X,Y) is always stable for every Y[X]. In this paper, we show that universally right-stable...

Universality, complexity and asymptotically uniformly smooth Banach spaces

Ryan M. Causey, Gilles Lancien (2023)

Commentationes Mathematicae Universitatis Carolinae

For 1 < p ≤ ∞ , we show the existence of a Banach space which is both injectively and surjectively universal for the class of all separable Banach spaces with an equivalent p -asymptotically uniformly smooth norm. We prove that this class is analytic complete in the class of separable Banach spaces. These results extend previous works by N. J. Kalton, D. Werner and O. Kurka in the case p = ∞ .

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Bruno de Mendonça Braga (0)

Annales de l’institut Fourier

Upper and lower estimates for Schauder frames and atomic decompositions

Kevin Beanland, Daniel Freeman, Rui Liu (2015)

Fundamenta Mathematicae

We prove that a Schauder frame for any separable Banach space is shrinking if and only if it has an associated space with a shrinking basis, and that a Schauder frame for any separable Banach space is shrinking and boundedly complete if and only if it has a reflexive associated space. To obtain these results, we prove that the upper and lower estimate theorems for finite-dimensional decompositions of Banach spaces can be extended and modified to Schauder frames. We show as well that if a separable...

Upper and lower estimates in Banach sequence spaces

Raquel Gonzalo (1995)

Commentationes Mathematicae Universitatis Carolinae

Here we study the existence of lower and upper ℓ p -estimates of sequences in some Banach sequence spaces. We also compute the sharp ℓ p estimates in their basis. Finally, we give some applications to weak sequential continuity of polynomials.

Upper estimates on self-perimeters of unit circles for gauges

Horst Martini, Anatoliy Shcherba (2016)

Colloquium Mathematicae

Let M² denote a Minkowski plane, i.e., an affine plane whose metric is a gauge induced by a compact convex figure B which, as a unit circle of M², is not necessarily centered at the origin. Hence the self-perimeter of B has two values depending on the orientation of measuring it. We prove that this self-perimeter of B is bounded from above by the four-fold self-diameter of B. In addition, we derive a related non-trivial result on Minkowski planes whose unit circles are quadrangles.

Using boundaries to find smooth norms

Victor Bible (2014)

Studia Mathematica

The aim of this paper is to present a tool used to show that certain Banach spaces can be endowed with C k smooth equivalent norms. The hypothesis uses particular countable decompositions of certain subsets of B X * , namely boundaries. Of interest is that the main result unifies two quite well known results. In the final section, some new corollaries are given.

Valdivia compacta and equivalent norms

Ondřej Kalenda (2000)

Studia Mathematica

We prove that the dual unit ball of a Banach space X is a Corson compactum provided that the dual unit ball with respect to every equivalent norm on X is a Valdivia compactum. As a corollary we show that the dual unit ball of a Banach space X of density ℵ 1 is Corson if (and only if) X has a projectional resolution of the identity with respect to every equivalent norm. These results answer questions asked by M. Fabian, G. Godefroy and V. Zizler and yield a converse to Amir-Lindenstrauss’ theorem.

Variations on Yano's extrapolation theorem.

David E. Edmunds, Miroslav Krbec (2005)

Revista Matemática Complutense

We give very short and transparent proofs of extrapolation theorems of Yano type in the framework of Lorentz spaces. The decomposition technique developed in Edmunds-Krbec (2000) enables us to obtain known and new results in a unified manner.

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