Subproduct systems with quantum group symmetry

We introduce a class of subproduct systems of finite dimensional Hilbert spaces whose fibers are defined by the Jones–Wenzl projections in Temperley–Lieb algebras. The quantum symmetries of a subclass of these systems are the free orthogonal quantum groups. For this subclass, we show that the corresponding Toeplitz algebras are nuclear C$^\*$-algebras that are $KK$-equivalent to $\mathbb C$ and obtain a complete list of generators and relations for them. We also show that their gauge-invariant subalgebras coincide with the algebras of functions on the end compactifications of the duals of the free orthogonal quantum groups. Along the way we prove a few general results on equivariant subproduct systems, in particular, on the behavior of the Toeplitz and Cuntz–Pimsner algebras under monoidal equivalence of quantum symmetry groups.