A note on positivity of elementary operators

Richard M. Timoney

Abstract

We show that operators on n ×n matrices which are representable in the form T(X) = \sum _{i = 1}^l a_i X b_i (for a_i and b_i n ×n matrices) and are k-positive for k = [ \sqrt{l} ] must be completely positive. As a consequence, elementary operators on a C*-algebra with minimal length l which are k-positive for k = [ \sum{l} ] must be completely positive.