[Paper Review] The Bochner-Schoenberg-Eberlein property for commutative Frechet algebras

Mitra Amiri, Ali Rejali|arXiv (Cornell University)|Dec 11, 2020

Advanced Operator Algebra Research20 references4 citations

TL;DR

This paper generalizes the Bochner-Schoenberg-Eberlein (BSE) property from commutative Banach algebras to commutative Fréchet algebras by introducing a family of seminorms on the space of BSE-functions. It establishes that the multiplier algebra embeds into the BSE space if and only if the algebra admits a bounded Δ-weak approximate identity, and proves that the BSE space is unital precisely when a certain supremum over seminorms is finite.

ABSTRACT

A class of commutative Banach algebras which satisfy a Bochner-Schoenberg-Eberlein-type inequality was introduced by Takahasi and Hatori. We generalize this property for the commutative Frechet algebra A. Furthermore, some of the main results in the class of Banach algebras, will be verified and generalized for the Frechet case.

Motivation & Objective

• To extend the Bochner-Schoenberg-Eberlein (BSE) property from commutative Banach algebras to the broader class of commutative Fréchet algebras.
• To define and study the space $ C_{\text{BSE}}(\Delta(\mathcal{A})) $ equipped with a family of continuous seminorms $ (r_\ell) $, making it a Fréchet algebra.
• To generalize key results from the Banach algebra setting—particularly the characterization of BSE-algebras via approximate identities—into the Fréchet algebra framework.
• To establish conditions under which $ C_{\text{BSE}}(\Delta(\mathcal{A})) $ is unital, linking this to the finiteness of a specific seminorm supremum $ \beta_M(\mathcal{A}) $.

Proposed method

• Introduce a family of seminorms $ (r_\ell) $ on $ C_{\text{BSE}}(\Delta(\mathcal{A})) $, turning it into a commutative semisimple Fréchet algebra.
• Define the BSE-norm $ \|\sigma\|_{\text{BSE}} $ as the infimum of constants $ \beta $ satisfying a pointwise inequality over finite sums of characters.
• Adapt the notion of bounded $ \Delta $-weak approximate identity from Banach algebras to Fréchet algebras using the topology of pointwise convergence on the Gelfand spectrum.
• Use the strong topology on the dual space $ \mathcal{A}^* $, generated by seminorms $ P_M $ for bounded sets $ M \subset \mathcal{A} $, to define the critical quantity $ \beta_M(\mathcal{A}) $.
• Establish the equivalence $ \widehat{M(\mathcal{A})} \subseteq C_{\text{BSE}}(\Delta(\mathcal{A})) \Leftrightarrow \mathcal{A} \text{ has a bounded } \Delta\text{-weak approximate identity} $ via approximation arguments.
• Prove that $ C_{\text{BSE}}(\Delta(\mathcal{A})) $ is unital if and only if $ \beta_M(\mathcal{A}) < \infty $ for some bounded $ M \subset \mathcal{A} $, using duality and norm estimates.

Experimental results

Research questions

• RQ1 How can the BSE property be generalized from commutative Banach algebras to commutative Fréchet algebras?
• RQ2 What is the appropriate topological structure on the space of BSE-functions in the Fréchet setting?
• RQ3 Does the characterization of BSE-algebras via the multiplier algebra embedding into the BSE space hold in the Fréchet algebra context?
• RQ4 Under what conditions is the space $ C_{\text{BSE}}(\Delta(\mathcal{A})) $ unital in the Fréchet algebra setting?
• RQ5 How does the finiteness of $ \beta_M(\mathcal{A}) $ relate to the existence of a bounded $ \Delta $-weak approximate identity?

Key findings

• The space $ C_{\text{BSE}}(\Delta(\mathcal{A})) $, equipped with the family of seminorms $ (r_\ell) $, forms a commutative semisimple Fréchet algebra.
• The multiplier algebra $ \widehat{M(\mathcal{A})} $ is contained in $ C_{\text{BSE}}(\Delta(\mathcal{A})) $ if and only if $ \mathcal{A} $ admits a bounded $ \Delta $-weak approximate identity.
• The space $ C_{\text{BSE}}(\Delta(\mathcal{A})) $ is unital if and only if $ \beta_M(\mathcal{A}) < \infty $ for some bounded set $ M \subset \mathcal{A} $, where $ \beta_M(\mathcal{A}) $ is defined as the supremum of $ |\sum c_i| $ over normalized finite sums in the dual seminorm topology.
• The constant function 1 belongs to $ C_{\text{BSE}}(\Delta(\mathcal{A})) $ precisely when $ \beta_M(\mathcal{A}) < \infty $, confirming unitality via duality.
• The proof relies on the strong topology on $ \mathcal{A}^* $, with seminorms $ P_M $, and the fact that $ P_M(\sigma) = \sup \left| \sum c_i \sigma(\varphi_i) \right| $ under the constraint $ P_M(\sum c_i \varphi_i) \leq 1 $.
• The equivalence between the existence of a bounded $ \Delta $-weak approximate identity and the inclusion $ \widehat{M(\mathcal{A})} \subseteq C_{\text{BSE}}(\Delta(\mathcal{A})) $ generalizes a classical result from Banach algebras to the Fréchet setting.