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

## TL;DR

This paper characterizes absolutely compatible pairs of elements within von Neumann algebras, revealing their structure and similarities to generic pairs of projections, thus advancing understanding of compatibility in operator algebras.

## Contribution

It provides a complete description of absolutely compatible pairs of strict elements in von Neumann algebras, highlighting their structural form and relation to projection pairs.

## Key findings

- Characterization of absolutely compatible pairs in von Neumann algebras
- Structural resemblance to generic pairs of projections
- Complete description of strict element pairs

## Abstract

Let $A$ be a unital C$^*$-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, $\vert a - b \vert + \vert 1_A - a - b \vert = 1_A.$ 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.