On the inclusion $\cO_2 \subset \cQ_2$

TL;DR

This paper proves that the inclusion of the Cuntz algebra  into the diadic -algebra  is both $C^*$-irreducible and rigid, leading to the isomorphism of their injective envelopes.

Contribution

It establishes the $C^*$-irreducibility and rigidity of the inclusion     , and shows their injective envelopes are $*$-isomorphic.

Findings

The inclusion      is $C^*$-irreducible.

The inclusion is rigid.

Injective envelopes of  and   are $*$-isomorphic.

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.