Hilbert norms for graded algebras

TL;DR

This paper investigates Hilbert norms on graded algebras, establishing their existence with continuous multiplication but also showing limitations in choosing submultiplicative norms equal to one on the unit.

Contribution

It provides the first construction of Hilbert norms on certain graded algebras that ensure continuous multiplication, highlighting inherent limitations.

Findings

Existence of Hilbert norms with continuous multiplication on some graded algebras

Such norms cannot be submultiplicative and equal to one on the unit

Highlights fundamental constraints in superanalysis contexts

Abstract

This paper presents a solution to a problem from superanalysis about the existence of Hilbert-Banach superalgebras. Two main results are derived: 1) There exist Hilbert norms on some graded algebras (infinite-dimensional superalgebras included) with respect to which the multiplication is continuous. 2) Such norms cannot be chosen to be submultiplicative and equal to one on the unit of the algebra.