Induced Quadratic Modules

Abstract

Positivity in *-algebras can be defined either algebraically, by quadratic modules, or analytically, by *-representations. By the induction procedure for *-representations, we can lift the analytical notion of positivity from a *-subalgebra to the entire *-algebra. The aim in this article is to define and study the induction procedure for quadratic modules. The main question is when a given quadratic module on the *-algebra is induced from its intersection with the *-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.

Key Words:

• Central simple algebra
• Galois extension
• Induced representation
• Involution
• Quadratic module
• Real field
• Weyl algebra

2010 Mathematics Subject Classification:

• 14P99
• 14A22
• 47L60
• 13J25
• 06F25
• 12J15

Notes

¹Here, ℕ₀ = {0, 1, 2,…} is the set of all nonnegative integers.

Communicated by A. Wadsworth.