Induced quadratic modules in $*$-algebras

TL;DR

This paper explores how to extend algebraic notions of positivity from subalgebras to larger *-algebras using an induction process, addressing complex questions about quadratic modules and their origins.

Contribution

It introduces and studies an induction procedure for quadratic modules in *-algebras, linking algebraic and analytical positivity concepts and addressing when modules are induced from subalgebras.

Findings

Induction procedure for quadratic modules is defined and analyzed.

Characterization of when quadratic modules are induced from subalgebras.

Results are obtained only in very special cases for the smallest quadratic module.

Abstract

Positivity in $\ast$-algebras can be defined either algebraically, by quadratic modules, or analytically, by $\ast$-representations. By the induction procedure for $\ast$-representations we can lift the analytical notion of positivity from a $\ast$-subalgebra to the entire $\ast$-algebra. The aim of this paper is to define and study the induction procedure for quadratic modules. The main question is when a given quadratic module on the $\ast$-algebra is induced from its intersection with the $\ast$-subalgebra. This question is very hard even for the smallest quadratic module (i.e. the set of all sums of hermitian squares) and will be answered only in very special cases.