Ultraproducts and matricial quantum Gromov Hausdorff completeness for C*-algebras

We study the Lip-normed C*-algebras introduced by M. Rieffel, showing that the family of equivalence classes up to isomorphism preserving the Lip-seminorm is not complete w.r.t. the complete 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 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, necessary and sufficient conditions are given for the existence, on the limiting operator system, of a C*-product structure inherited from the approximating C*-algebras. Such conditions can be considered as a generalisation of the f-Leibnitz conditions considered by Li and Kerr. Furthermore, it is shown that our conditions are 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 from the approximating C*-algebras.