Information, Geometry, and Physics Seminar

Let Φ be a unital completely positive map on the space of operators on some Hilbert space. We assume that Φ is almost idempotent, namely, ‖Φ2−Φ‖cb≤η , and construct a corresponding "ε-C∗ algebra" for ε=O(η) .This type of structure has the axioms of a unital C∗ algebra but the associativity and other axioms involving the multiplication and the unit hold up to ε. We further prove that any finite-dimensional ε-C∗ algebra is O(ε)-isomorphic to a genuine C∗ algebra. These bounds are universal, i.e. do not depend on the dimensionality or other parameters.