Commutativity of operators and characterization of traces on C*-algebras

Abstract

79 This paper continues the author’s study begun in [1, 2]; we retain the notation and terminology used there. In Section 2, we give new criteria for the com� mutativity of a nonnegative operator and a projection in terms of operator inequalities. We show that, in the general case, it is impossible to replace the projection in these inequalities by any nonnegative operator so as to preserve the commutativity of operators. We also obtain new commutativity criteria for projections in terms of operator inequalities. In Section 3, we obtain a characterization of traces in the class of all positive normal functionals on the von Neumann algebra by using the operator inequali� ties from Section 2. It is shown that not each charac� teristic which distinguishes traces among positive nor� mal functionals carries over to normal weights (see Example 3.1). This answers a question which Victor Kaftal (Cincinnati University, U.S.A.) asked the author at the international conference Operator The� ory’23 (Romania, Timisoara, 2010, June 30). We also give a characterization of traces in the class of all weights on von Neumann algebra. In Section 4, we obtain a characterization of traces in the class of all positive functionals on the C*�algebra in terms of operator inequalities. Moreover, we prove new criteria for the commutativity of C*�algebras. An information about other characterizations of traces and commuta� tivity criteria for C*�algebras can be found in the author’s survey [3].