Quantum Gromov-Hausdorff Convergence for Extensions of $C^*$-Algebras

We study Toeplitz type $C^*$-algebraic extensions of unital $C^*$-algebras by stable ideals, from the perspective of noncommutative metric geometry. Using the spectral metric space construction of Hawkins and Zacharias (Comm. Math. Phys. 350 (2017), 475-506), we analyze the interaction of these extensions with the quantum Gromov-Hausdorff distance. We show that complete sub-operator systems of the quotient, or of the unital algebra underlying the stable ideal, canonically determine complete sub-operator systems of the extension. We introduce the notions of unital 2-contractive approximation and its Toeplitz type refinement as our key approximation tools. Our main results show that if a sequence of complete sub-operator systems of the unital algebra underlying the stable ideal converges in the quantum Gromov-Hausdorff distance under the unital 2-contractive approximation condition and a compatibility condition on the quotient, then the corresponding sequence in the extension also converges. An analogous statement holds from the quotient to the extension under the 2-contractive Toeplitz type refinement condition.