Extensions of Operator Systems, Hyperrigidity and Quantum Double Suspension

We introduce a class of extensions of unital operator systems satisfying a certain Toeplitz type property. We show that these extensions induce corresponding extensions of the $C^{\ast }$-algebras they generate. We prove that hyperrigidity is preserved under these extensions, yielding new examples through iteration. We introduce the quantum double suspension of operator systems as a special class of these extensions and show that its iterates yield the Hong-Szymański quantum double suspensions at the $C^{\ast }$-algebra level, encompassing, in particular, the Vaksman-Soibelman quantum spheres. We construct hyperrigid operator systems underlying these quantum spheres. As a further application, we show that every counterexample to Arveson’s hyperrigidity conjecture gives rise to a sequence of counterexamples. We identify the $C^{\ast }$-envelopes of these extensions and use this identification to obtain a $K$-theoretic condition ensuring that each such sequence contains infinitely many pairwise non-isomorphic counterexamples. We also observe that the validity of the conjecture in the commutative setting is linked to its validity for an infinite family of noncommutative operator systems obtained through successive extensions. Finally, we compute the propagation number, an invariant of operator systems introduced by Connes and van Suijlekom, for these extensions.