Cones and projections in corona relative commutants

We study the relative commutants of separable subalgebras of corona C*-algebras. If D is nonzero, simple, sigma-unital and nonunital, then every such commutant in Q(D)=M(D)/D contains the cone C_0((0,1]) tensor B over every separable C*-algebra B. These commutants are consequently nonexact. If the unital algebra generated by the probe has a nonzero exact quotient, its commutant contains a nonzero, possibly nonunital copy of O_2. For a nonempty compact metrizable space X without isolated points, we construct a singly generated unital subalgebra of Q(C(X) tensor K) whose commutant has exactly the scalar projections given by clopen subsets of X. In particular, the interval admits such a commutant with no projections other than 0 and 1. The latter example admits only scalar-unitary representations of C_2*C_3, whereas every commutant in the simple-base case admits uniformly bounded representations of the free group F_2 with bound arbitrarily close to one that cannot be unitarized within the commutant. Paper source and revision history: https://github.com/eulogica/cones-projections-corona