Von-Neumann finiteness and reversibility in some classes of non-associative algebras

TL;DR

This paper explores conditions for von-Neumann finiteness and reversibility in certain non-associative algebras, establishing finiteness in finite-dimensional alternative algebras and those from Cayley-Dickson doubling, with criteria based on subalgebra isomorphisms.

Contribution

It provides new criteria for von-Neumann finiteness and reversibility in involutive non-associative algebras, including classifications based on 3-dimensional subalgebras.

Findings

Finite-dimensional alternative algebras are von-Neumann finite.

Algebras from Cayley-Dickson doubling are von-Neumann finite.

Criteria for involutive algebra properties are given in terms of subalgebra isomorphisms.

Abstract

We investigate criteria for von-Neumann finiteness and reversibility in some classes of non-associative algebras. We show that all finite-dimensional alternative algebras, as well as all algebras obtained from the real numbers via the standard Cayley-Dickson doubling process, are von-Neumann finite. Precise criteria for von-Neumann finiteness and reversibility of involutive algebras are given, in terms of isomorphism types of their 3-dimensional subalgebras.