Just-infinite Jordan Banach algebras

TL;DR

This paper introduces and studies the concept of just-infinite JB-algebras, exploring their structure, classification, and relation to just-infinite C*-algebras, revealing that they are either spin factors or related to specific subalgebras of C*-algebras.

Contribution

It defines just-infinite JB-algebras, analyzes their structure, and establishes their connection to just-infinite C*-algebras, providing a classification framework.

Findings

Any just-infinite JB-algebra is either an infinite-dimensional spin factor or related to subalgebras of a C*-algebra.

Established a correspondence between just-infinite C*-algebras and their Jordan algebras of self-adjoint elements.

Connected the properties of just-infinite JB-algebras to the structure of their parent C*-algebras.

Abstract

By analogy with the well-established notions of just-infinite groups and just-infinite algebras, in particular $C^*$-algebras, we initiate a study of just-infinite $JB$-algebras, i.e. infinite dimensional $JB$-algebras for which all proper quotients are finite dimensional. We investigate the connections between a just-infinite $C^*$-algebra $A$ and its Jordan algebra $H(A,*)$ of self-adjoint elements. We also show that any just-infinite $JB$-algebra $J$ either is a infinite-dimensional spin factor or there exists a $C^*$-algebra $A$ and just-infinite norm-closed real $*$-subalgebras $A_1$ and $A_2$ of $A$ such that $H(A_1,*)\unlhd J \subseteq H(A_2,*).$