Almost-idempotent quantum channels and approximate $C^*$-algebras
Almost-idempotent quantum channels and approximate $C^*$-algebras
Let $\Phi$ be a unital completely positive map on the space of operators on some Hilbert space. We assume that $\Phi$ is almost idempotent, namely, $\|\Phi^2-\Phi\|_{\mathrm{cb}} \le\eta$, and construct a corresponding "$\varepsilon$-$C^*$ algebra" for $\varepsilon=O(\eta)$. This type of structure has the axioms of a unital $C^*$ algebra but the associativity …