Absolutely compatible pairs in a von Neumann algebra

Abstract

Let a, b be elements in a unital C * -algebra with 0 ≤ a, b ≤ 1. The element a is absolutely compatible with b if |a − b| + |1 − a − b| = 1.In this note we find some technical characterizations of absolutely compatible pairs in an arbitrary von Neumann algebra. These characterizations are applied to measure how close is a pair of absolute compatible positive elements in the closed unit ball from being orthogonal or commutative. In the case of 2 by 2 matrices the results offer a geometric interpretation in terms of an ellipsoid determined by one of the points. The conclusions for 2 by 2 matrices are also applied to describe absolutely compatible pairs of positive elements in the closed unit ball of Mn.1 2 for the absolute value of a. Several attempts to establish a non-commutative version of the celebrated Kakutani's theorem [3], which characterizes those Banach lattices which are lattice isomorphic to the space C(Ω), of all continuous functions on a compact Hausdorff space Ω, have been pursued in recent years (cf. [4, 5, 6]). As in many previous forerunners, like the representation theory, published by Stone in [10], which characterizes C(Ω) in terms of order and its ring properties, a non-commutative Kakutani's theorem will necessarily rely on the notions of orthogonality, absolute value and order. Some discoveries have been found within this non-commutative program, for example, it is shown in [7, Proposition 4.9] that if a is an arbitrary positive element in the closed unit ball, B A , of a unital C * -algebra A and p is a projection in A, then |p − a| + |1 − p − a| = 1 if and only if a and p commute. Furthermore, two positive elements a and b in B A are orthogonal if, and only if, a + b ≤ 1 and |a − b| + |1 − a − b| = 1. The second condition gives rise to a strictly weaker notion than the usual orthogonality. Accordingly to the notation in 1991 Mathematics Subject Classification. Primary 46L10; Secondary 46B40 46L05.