The Restricted Weyl Group of the Cuntz Algebra and Shift Endomorphisms

TL;DR

This paper characterizes a specific subgroup of automorphisms of the Cuntz algebra O_n that preserve certain subalgebras, linking them to homeomorphisms of the one-sided shift space and describing their structure in relation to shift automorphisms.

Contribution

It establishes an isomorphism between a subgroup of automorphisms of O_n and homeomorphisms of the shift space that commute with the shift, clarifying their relation to the outer automorphism group.

Findings

Automorphisms fixing the diagonal MASA correspond to shift space homeomorphisms.

The subgroup embeds into the quotient of the automorphism group of the two-sided shift.

For prime n, the embedding is an isomorphism.

Abstract

It is shown that, modulo the automorphisms which fix the canonical diagonal MASA point-wise, the group of those automorphisms of the Cuntz algebra O_n which globally preserve both the diagonal and the core UHF-subalgebra is isomorphic, via restriction, with the group of those homeomorphisms of the full one-sided n-shift space which eventually commute along with their inverses with the shift transformation. The image of this group in the outer automorphism group of O_n can be embedded into the quotient of the automorphism group of the full two-sided n-shift by its center, generated by the shift. If n is prime then this embedding is an isomorphism.