On the inclusion $\cO_2 \subset \cQ_2$

TL;DR

The paper studies the canonical inclusion $O_2 \subset Q_2$, showing it is $C^*$-irreducible and rigid. By using the universal presentations of $O_2$ and $Q_2$, the Cartan diagonal subalgebra $D_2$, and the framework of pseudo-expectations and injective envelopes, the authors establish that the inclusion has no nontrivial intermediate algebras and that the only unital completely positive map on $Q_2$ fixing $O_2$ is the identity. They further show that the injective envelopes $I(O_2)$ and $I(Q_2)$ are $*$-isomorphic, extending the analysis to related square and core algebras and supporting the conjecture that similar results hold for $O_n \subset Q_n$ for $n>2$. The findings provide a concrete example of a $C^*$-irreducible, rigid inclusion with identical injective envelopes, with potential implications for Cartan subalgebras and dynamical constructions in diadic and Cuntz algebras.

Abstract

The diadic $C^*$-algebra $\cQ_2$ contains canonically a copy of the Cuntz algebra $\cO_2$. It is shown that the inclusion $\cO_2 \subset \cQ_2$ is $C^*$-irreducible and rigid. It follows that the injective envelopes of these two $C^*$-algebras are $*$-isomorphic.

Theorems & Definitions (12)

• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Corollary 2.3
• proof
• Remark 2.4
• Proposition 2.5
• proof
• Corollary 2.6