Absolutely compatible pair of elements in a von Neumann algebra-II

Abstract

Let $A$ be a unital C${}^{\ast }$-algebra with unity $1_A$. A pair of elements $0\le a,b\le 1_A$ in $A$ is said to be \emph{absolutely compatible} if, $|a-b|+|1_A-a-b|=1_A.$ In this paper we provide a complete description of absolutely compatible pair of strict elements in a von Neumann algebra. The end form of such a pair has a striking resemblance with that of a `generic pair' of projections on a complex Hilbert space introduced by Halmos.

7 pages