# Absolutely compatibility in $\\mathbb{M}\_2$ and ellipticity

Source: [Absolutely compatibility in $\\mathbb{M}\_2$ and ellipticity](<https://www.sota2.com/research/paper/68fa554a1e209131ce992edf>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T13:30:44.305Z

Authors: Nabin K. Jana, Anil K. Karn

Published: 2018-04-17T15:36:25.000Z

## Abstract

Let $a,b$ be elements in a unital C$^\*$-algebra with $0\\leq a,b\\leq 1$. The element $a$ is absolutely compatible with $b$ if $$\\vert a - b \\vert + \\vert 1 - a - b \\vert = 1.$$ In this note, we describe a complete list of absolutely compatible pairs in $\\mathbb{M}\_2$. We note that non-commutative pairs exhibit an elliptic behaviour.

## Recorded benchmark results

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1804.06325>)

[PDF](<https://arxiv.org/pdf/1804.06325.pdf>)