The problem of completeness for Gromov–Hausdorff metrics on C*-algebras

It is proved that the family of equivalence classes of Lip-normed C*-algebras introduced by M. Rieffel, up to complete order isomorphisms preserving the Lip-seminorm, is not complete w.r.t. the matricial quantum Gromov–Hausdorff distance introduced by D. Kerr. This is shown by exhibiting a Cauchy sequence whose limit, which always exists as an operator system, is not completely order isomorphic to any C*-algebra.Conditions ensuring the existence of a C*-structure on the limit are considered, making use of the notion of ultraproduct. More precisely, a necessary and sufficient condition is given for the existence, on the limiting operator system, of a C*-product structure inherited from the approximating C*-algebras. Such condition can be considered as a generalisation of the f-Leibniz conditions introduced by Kerr and Li. Furthermore, it is shown that our condition is not necessary for the existence of a C*-structure tout court, namely there are cases in which the limit is a C*-algebra, but the C*-structure is not inherited.