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An order-theoretic characterization of JB-algebras

Mark Roelands, Samuel Tiersma

Abstract

We give an order-theoretic characterization of the JB-algebras among the complete order unit spaces in terms of the existence of an order-anti-automorphism of the interior of the cone that is homogeneous of degree -1. More geometrically, we characterize JB-algebras as those complete order unit spaces for which the interior of the cone is a symmetric Banach--Finsler manifold under Thompson's metric. Furthermore, we show that two order unit spaces are isomorphic if there exists a gauge-reversing bijection between them, thus answering a question raised by Noll--Schäfer. These results have previously been established for finite-dimensional resp. reflexive order unit spaces by Walsh and Lemmens--R.--Wortel.

An order-theoretic characterization of JB-algebras

Abstract

We give an order-theoretic characterization of the JB-algebras among the complete order unit spaces in terms of the existence of an order-anti-automorphism of the interior of the cone that is homogeneous of degree -1. More geometrically, we characterize JB-algebras as those complete order unit spaces for which the interior of the cone is a symmetric Banach--Finsler manifold under Thompson's metric. Furthermore, we show that two order unit spaces are isomorphic if there exists a gauge-reversing bijection between them, thus answering a question raised by Noll--Schäfer. These results have previously been established for finite-dimensional resp. reflexive order unit spaces by Walsh and Lemmens--R.--Wortel.

Paper Structure

This paper contains 31 sections, 91 theorems, 208 equations.

Table of Contents

  1. Introduction
  2. Functional-analytic perspective.
  3. Differential-geometric perspective.
  4. Algebraic perspective.
  5. Preliminaries
  6. Differentiable maps
  7. Analytic maps
  8. Semi-continuous functions
  9. Order unit spaces
  10. Gauges and Thompson's metric
  11. Gauge-reversing bijections
  12. Jordan algebras
  13. JB-algebras
  14. Extended cones of semicontinuous affine functions
  15. Extension of the cones
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $(V, V_+, v)$ be a complete order unit space. Then there exists a structure of Jordan algebra on $V$ such that $(V, \lVert\cdot\rVert_v)$ is a JB-algebra with unit element $v$ and cone of squares $V_+$ if and only if there exists a gauge-reversing bijection on $V^\circ_+$.

Theorems & Definitions (207)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Hua's identity
  • Theorem 1.4
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • ...and 197 more