The ternary subspace and symmetric part of an operator space

Abstract

In 2003, V. I. Paulsen and I denned the ternary subspace of an operator space as the intersection of the space and the adjoint of its quasi-multiplier space. Recently, M. Neal and B. Russo defined the completely symmetric part of an operator space by considering the symmetric part of the matrix of infinite size w i t h entries in the operator space, and posed the question: Under what conditions does it consist of the adjoint of quasi-multipliers? I give a partial answer to this question revealing the relationship between the ternary subspace and the completely symmetric part.