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Model completeness and quantifier elimination for (ordered) central simple algebras with involution

Vincent Astier

TL;DR

This work develops a model-theoretic framework for ordered central simple algebras with involution over real closed fields. It proves that CSA$_m$ and CSA-I$_m$ are model-complete when the center (or base field) is real closed, and establishes quantifier elimination for matrix-algebra theories with involution by introducing positive cones and type-specification, enabling controlled amalgamation. The authors define precise matrix-basis axioms, connect positivity to hermitian squares, and show QE for theories like $\text{OCSA-OI}_{m,rcf}^+$ and its variants. A key outcome is a correspondence between positive cones and morphisms into models of the ordered theory, integrating order-theoretic and algebraic structure. The results lay groundwork for decidability questions and real-algebraic methods in noncommutative settings, with potential applications to signatures of hermitian forms, unitary similarity, and the interplay between orderings and involutions.

Abstract

We show that the theories of some (ordered) central simple algebras with involution over real closed fields are model-complete or admit quantifier elimination, and characterize positive cones in terms of morphisms into models of some of these theories.

Model completeness and quantifier elimination for (ordered) central simple algebras with involution

TL;DR

This work develops a model-theoretic framework for ordered central simple algebras with involution over real closed fields. It proves that CSA and CSA-I are model-complete when the center (or base field) is real closed, and establishes quantifier elimination for matrix-algebra theories with involution by introducing positive cones and type-specification, enabling controlled amalgamation. The authors define precise matrix-basis axioms, connect positivity to hermitian squares, and show QE for theories like and its variants. A key outcome is a correspondence between positive cones and morphisms into models of the ordered theory, integrating order-theoretic and algebraic structure. The results lay groundwork for decidability questions and real-algebraic methods in noncommutative settings, with potential applications to signatures of hermitian forms, unitary similarity, and the interplay between orderings and involutions.

Abstract

We show that the theories of some (ordered) central simple algebras with involution over real closed fields are model-complete or admit quantifier elimination, and characterize positive cones in terms of morphisms into models of some of these theories.

Paper Structure

This paper contains 13 sections, 29 theorems, 42 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Algebras with involution
  3. Model-theoretic notation
  4. Axiomatization of finite-dimensional central simple algebras
  5. The reduced trace, the $*$ operation, and words of matrices
  6. Model completeness
  7. Matrix bases
  8. Model completeness
  9. Quantifier elimination
  10. Positive cones and positive involutions
  11. Model-completeness
  12. Quantifier elimination with an involution of specified type
  13. Correspondence between positive cones and morphisms

Key Result

Proposition 1.1

If $\sigma$ and $\gamma$ are two $F$-linear involutions on $A$ and are of the same kind, then there is $a \in A^\times$ such that $\sigma = \mathop{\mathrm{Int}}\nolimits(a) \circ \gamma$.

Theorems & Definitions (66)

  • Proposition 1.1: BOI
  • Remark 1.2
  • Lemma 1.3
  • proof
  • Corollary 1.4
  • Remark 1.5
  • Lemma 1.6
  • proof
  • Lemma 1.7
  • proof
  • ...and 56 more