Automorphisms of the UHF algebra that do not extend to the Cuntz algebra

TL;DR

The paper investigates conditions under which automorphisms of the UHF subalgebra of the Cuntz algebra can or cannot be extended to automorphisms of the entire Cuntz algebra, providing specific examples and criteria.

Contribution

It offers necessary and sufficient conditions for extending automorphisms from the UHF subalgebra to the Cuntz algebra, including automorphisms of the diagonal MASA.

Findings

Automorphisms of the UHF subalgebra do not always extend to the Cuntz algebra.

Criteria for extension involve properties of infinite tensor products of automorphisms.

Existence of automorphisms of the diagonal that are not extendable to endomorphisms of O_n.

Abstract

Automorphisms of the canonical core UHF-subalgebra F_n of the Cuntz algebra O_n do not necessarily extend to automorphisms of O_n. Simple examples are discussed within the family of infinite tensor products of (inner) automorphisms of the matrix algebras M_n. In that case, necessary and sufficient conditions for the extension property are presented. It is also addressed the problem of extending to O_n the automorphisms of the diagonal D_n, which is a regular MASA with Cantor spectrum. In particular, it is shown the existence of product-type automorphisms of D_n that are not extensible to (possibly proper) endomorphisms of O_n.