IUMJ

Title: On invariant MASAs for endomorphisms of the Cuntz algebras

Authors: Jeong Hee Hong, Adam Skalski and Wojciech Szymanski

Issue: Volume 59 (2010), Issue 6, 1873-1892

Abstract:

The problem of existence of standard (i.e., product-type) invariant MASAs for endomorphisms of the Cuntz algebra $\mathcal{O}_n$ is studied. In particular, endomorphisms which preserve the canonical diagonal MASA $\mathcal{D}_n$ are investigated. Conditions on a unitary $w\in\mathcal{U}(\mathcal{O}_n)$ equivalent to the fact that the corresponding endomorphism $\lambda_w$ preserves $\mathcal{D}_n$ are found, and it is shown that they may be satisfied by unitaries which do not normalize $\mathcal{D}_n$. 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 $\mathcal{O}_n$ given by Izumi and related to sector theory are discussed and it is shown that they lead to an endomorphism of $\mathcal{O}_2$ associated to a matrix unitary which does not preserve any standard MASA.