Discrete inclusions from Cuntz-Pimsner algebras

We show that the core inclusion arising from a Cuntz-Pimsner algebra generated by a full, faithful and dualizable correspondence is C*-discrete, and express it as a crossed-product by an action of a tensor category. In particular, we show <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper U sans-serif upper H sans-serif upper F Subscript n Sub Superscript normal infinity Baseline subset-of script upper O Subscript n"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">U</mml:mi> <mml:mi mathvariant="sans-serif">H</mml:mi> <mml:mi mathvariant="sans-serif">F</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>n</mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msup> </mml:mrow> </mml:msub> <mml:mo> ⊂ </mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {UHF}_{n^\infty }\subset \mathcal {O}_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the inclusion of the fixed-point algebra under the gauge symmetry of the Cuntz algebra, is C*-discrete. We describe the dualizable bimodules appearing under this inclusion, including their semisimple decompositions and fusion rules, their Watatani indices and Pimsner-Popa bases, and their sets of cyclic algebraic generators.