Smith–Ward problem for simultaneous preservation of all matrix ranges of a single operator

Determine whether, for every element T in the Calkin algebra Q(H) on a separable infinite-dimensional Hilbert space H, there exists an operator S in B(H) such that the canonical quotient map π:B(H)→Q(H) satisfies π(S)=T and, for all n∈N, the n-th matrix ranges coincide: W^n(S)=W^n(T). Here, W^n(X) denotes the set {Φ(X): Φ is a unital completely positive map into M_n}.

Background

The Smith–Ward Theorem shows that for the canonical quotient map π:B(H)→Q(H) and any operator T∈B(H), there exists a compact operator K such that the n-th matrix ranges are preserved under the compact perturbation T+K for all 1≤n≤N, where N is fixed. Li–Paulsen–Poon extended this to joint matrix ranges of m-tuples of self-adjoint operators, again up to a fixed N.

The unresolved case asks whether one can drop the dependence on N when m=1, i.e., for a single operator, and preserve all matrix ranges simultaneously. This is known as the Smith–Ward problem. It is equivalent to whether every three-dimensional operator system has the lifting property, linking matrix range preservation to operator system lifting theory.