On Fr\'echet-Hilbert Algebras

TL;DR

This paper explores Hilbert algebras with additional Fréchet topologies, extending their structure via duality, and constructs large multiplier-type involutive algebras applicable in quantization contexts.

Contribution

It introduces new extensions of Hilbert algebras with Fréchet topology, including optimal multiplier-type involutive algebras for practical applications.

Findings

Constructed large multiplier-type involutive algebras

Extended algebraic structures using duality techniques

Applicable to quantization involving square-integrable operator families

Abstract

We consider Hilbert algebras with a supplementary Fr\'echet topology and get various extensions of the algebraic structure by using duality techniques. In particular we obtain optimal multiplier-type involutive algebras, which in applications are large enough to be of significant practical use. The setting covers many situations arising from quantization rules, as those involving square-integrable families of bounded operators.