The natural inclusion of the Cuntz algebra O2 in the diadic C*-algebra Q2 is C*-irreducible and rigid, so their injective envelopes are *-isomorphic.
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On the inclusion $\cO_2 \subset \cQ_2$
The natural inclusion of the Cuntz algebra O2 in the diadic C*-algebra Q2 is C*-irreducible and rigid, so their injective envelopes are *-isomorphic.