If $A$ is a abelian $C^∗$-algebra and $a,b$ are elements in $A$ such that $0≤a≤1,0≤b≤1$ then $0≤\|a-b \|≤1$.
My problem is:"Does the same hold if $A$ is not abelian?"
2026-04-04 15:13:03.1775315583
positive elements and norm
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Yes. You have $$ -1\leq -b\leq a-b\leq a\leq1. $$