On the outer automorphism groups of triangular alternation limit algebras

TL;DR

This paper characterizes the outer automorphism group of the alternation limit algebra, revealing it as an abelian group determined by prime divisors of the defining sequences, linking algebraic automorphisms to number-theoretic properties.

Contribution

It provides a complete description of the outer automorphism group of the alternation limit algebra in terms of prime divisors of the sequences defining its structure.

Findings

Outer automorphism group is isomorphic to a7^d, where d counts primes dividing infinitely many terms.

The automorphism group relates to the automorphisms of the fundamental relation of the algebra.

The quotient of the isometric automorphism group by approximately inner automorphisms is abelian.

Abstract

Let $A$ denote the alternation limit algebra, studied by Hopenwasser and Power, and by Poon, which is the closed direct limit of upper triangular matrix algebras determined by refinement embeddings of multiplicity $r_k$ and standard embeddings of multiplicity $s_k$. It is shown that the quotient of the isometric automorphism group by the approximately inner automorphisms is the abelian group $ \ZZ ^d$ where $d$ is the number of primes that are divisors of infinitely many terms of each of the sequences $(r_k)$ and $(s_k)$. This group is also the group of automorphisms of the fundamental relation of $A$.