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Just-infinite Jordan Banach algebras

Victor Zhelyabin, Andrey Mamontov

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 $H(A_1,*)\unlhd J \subseteq H(A_2,*).$

Just-infinite Jordan Banach algebras

Abstract

By analogy with the well-established notions of just-infinite groups and just-infinite algebras, in particular -algebras, we initiate a study of just-infinite -algebras, i.e. infinite dimensional -algebras for which all proper quotients are finite dimensional. We investigate the connections between a just-infinite -algebra and its Jordan algebra of self-adjoint elements. We also show that any just-infinite -algebra either is a infinite-dimensional spin factor or there exists a -algebra and just-infinite norm-closed real -subalgebras and of such that
Paper Structure (3 sections, 23 theorems, 57 equations)

This paper contains 3 sections, 23 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Jordan algebras of self-adjoint elements of just-infinite $C^*$-algebras
  3. The structure of just-infinite $JB$-algebras

Key Result

Lemma 1

Let $A$ be a $C^*$-algebra and let $I$ be a closed ideal of the $\mathbb{R}$-algebra $A$. Then $I$ is an ideal of the $C^*$-algebra $A$.

Theorems & Definitions (28)

  • Lemma 1
  • Lemma 2
  • Definition 1
  • Corollary 1
  • Example 1
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • ...and 18 more