Open Access
Submultiplicative norms in CC(K) spaces with applications to uniform algebras
(Springer, 2024) Albiac Alesanco, Fernando José; Blasco, Óscar; Briem, E.; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
Certain algebra norms on the algebra of functions on a two-point set are defined and used to construct a new family of parametric algebra norms on the space CC(K) of continuous complex-valued functions on a compact Hausdorff space. As a by-product of our work we transfer our construction to uniform algebras to obtain a new collection of norms with special properties.
Open Access
On the norm-preservation of squares in real algebra representation
(Springer, 2021) Albiac Alesanco, Fernando José; Blasco, Óscar; Briem, E.; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
One of the main results of the article Gelfand theory for real Banach algebras, recently published in [Rev R Acad Cienc Exactas Fís Nat Ser A Mat RACSAM 114(4):163, 2020] is Proposition 4.1, which establishes that the norm inequality ||a2||≤ ||a2+ b2|| for a, b∈ A is sufficient for a commutative real Banach algebra A with a unit to be isomorphic to the space CR(K) of continuous real-valued functions on a compact Hausdorff space K. Moreover, in this proposition is also shown that if the above condition (which involves all the operations of the algebra) holds, then the real-algebra isomorphism given by the Gelfand transform preserves the norm of squares. A very natural question springing from the above-mentioned result is whether an isomorphism of A onto CR(K) is always norm-preserving of squares. This note is devoted to providing a negative answer to this problem. To that end, we construct algebra norms on spaces CR(K) which are (1 + ϵ) -equivalent to the sup-norm and with the norm of the identity function equal to 1, where the norm of every nonconstant function is different from the standard sup-norm. We also provide examples of two-dimensional normed real algebras A where ||a2|| ≤ k|| a2+ b2|| for all a, b∈ A, for some k>1 , but the inequality fails for k= 1.
Open Access
Counterexamples in isometric theory of symmetric and greedy bases
(Elsevier, 2024) Albiac Alesanco, Fernando José; Ansorena, José L.; Blasco, Óscar; Chu, Hùng Việt; Oikhberg, Timur; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
We continue the study initiated in Albiac and Wojtaszczyk (2006) of properties related to greedy bases in the case when the constants involved are sharp, i.e., in the case when they are equal to 1. Our main goal here is to provide an example of a Banach space with a basis that satisfies Property (A) but fails to be 1-suppression unconditional, thus settling Problem 4.4 from Albiac and Ansorena (2017). In particular, our construction demonstrates that bases with Property (A) need not be 1-greedy even with the additional assumption that they are unconditional and symmetric. We also exhibit a finite-dimensional counterpart of this example, and show that, at least in the finite-dimensional setting, Property (A) does not pass to the dual. As a by-product of our arguments, we prove that a symmetric basis is unconditional if and only if it is total, thus generalizing the well-known result that symmetric Schauder bases are unconditional.

Albiac Alesanco, Fernando José

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