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.