On Invariant MASAs for Endomorphisms of the Cuntz Algebras

TL;DR

This paper investigates conditions under which endomorphisms of the Cuntz algebra O_n preserve invariant MASAs, characterizing unitaries involved and analyzing examples related to sector theory and finite-index endomorphisms.

Contribution

It provides new criteria for unitaries to preserve standard MASAs in O_n and characterizes endomorphisms that leave all standard MASAs invariant, expanding understanding of their structure.

Findings

Unitaries preserving D_n are characterized by specific conditions.

Some unitaries do not normalize D_n but still preserve it under endomorphisms.

Certain finite-index endomorphisms do not preserve any standard MASA.

Abstract

The problem of existence of standard (i.e. product-type) invariant MASAs for endomorphisms of the Cuntz algebra O_n is studied. In particular endomorphisms which preserve the canonical diagonal MASA D_n are investigated. Conditions on a unitary in O_n equivalent to the fact that the corresponding endomorphism preserves D_n are found, and it is shown that they may be satisfied by unitaries which do not normalize D_n. Unitaries giving rise to endomorphisms which leave all standard MASAs invariant and have identical actions on them are characterized. Finally some properties of examples of finite-index endomorphisms of O_n given by Izumi and related to sector theory are discussed and it is shown that they lead to an endomorphism of O_2 associated to a matrix unitary which does not preserve any standard MASA.