Alexei Kitaev (May 07 2024).
Abstract: 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 and other axioms involving the multiplication and the unit hold up to $\varepsilon$. We further prove that any finite-dimensional $\varepsilon$-$C^$ algebra is $O(\varepsilon)$-isomorphic to a genuine $C^$ algebra. These bounds are universal, i.e.\ do not depend on the dimensionality or other parameters.