A NEW PROOF FOR THE COMPLETE POSITIVITY OF THE BOOLEAN PRODUCT OF COMPLETELY POSITIVE MAPS BETWEEN C*-ALGEBRAS

The Boolean product of linear functionals on algebras is defined on the associated universal free product algebra without unit, and on involutive algebras it preserves the positivity (see, e.g., [15]). This product originates in M. Bozejko’s investigations on positive definite functions on the free group(see, e.g., [5]) and corresponds to the Boolean independence and partial cumulants going back to W. von Waldenfels’ work on the pressure broadening of spectral lines (see, e.g., [20]). This Boolean product and the involved independence are fundamental in the so-called Boolean quantum probability theory and related topics (see, e.g., [17, 2, 1, 3, 8, 10]). This theory is one of the three noncommutative probability theories (the other being R.L.Hudson’s Boson or Fermion probability theory and D.V. Voiculescu’s free probability theory) issued from an associative product which does not depend on the order of its factors and fulfills a universal rule for mixed moments ( according to R. Speicher’s answer[16] to M. Schurmann’s conjecture [15] on the universal products of *-algebraic probability spaces). In [9], we considered the Boolean product for linear maps between algebras and showed, by a direct proof, it preserves the complete positivity in C*-algebraic setting. In this Note, we give a new proof of the same fact, inspired from that in [6] concerning the positivity of the conditionally free product of positive linear functionals on involutive algebras.