A Three-Dimensional Operator System without the Smith--Ward Property
arXiv:2607.04274
The paper builds a three‑dimensional hyperrigid operator system inside a matrix amplification of the reduced C*-algebra of the free group, whose identity map lacks a unital completely positive lift, giving the first three‑dimensional counterexample to the Smith–Ward problem and showing its dual is not exact.
Abstract
Harris recently showed that a non-liftable injective representation into the Calkin algebra gives explicit four-dimensional operator systems in the Calkin algebra without the lifting property, and hence a counterexamples to the generalized Smith--Ward problem for four-dimensional operator systems. The main obstruction also appears in an earlier work by Paulsen on this problem. We isolate the relevant part of this argument and replace the four-dimensional operator system by a three-dimensional hyperrigid operator system inside a matrix amplification of \[ C_r^*(\F_2). \] The resulting Calkin subsystem is of the form span, where and are selfadjoint operators, and the identity map on this operator system has no unital completely positive lift. Equivalently, the operator gives a counterexample to the Smith--Ward problem. By a result of Kavruk, the dual of this operator system fails to be exact, and hence is the first example of a three-dimensional operator system that is not exact.
9 pages; replaced S in Corollary 4.3 with its dual; added keywords and fundings