A Projection Characterization and Symmetry Bootstrap for Elements of a von Neumann Algebra that are Nearby Commuting Elements

We define a symmetry map $\varphi$ on a unital $C^\ast$-algebra $\mathcal A$ to be an $\mathbb{R}$-linear map on $\mathcal A$ that generalizes transformations on matrices like: transpose, adjoint, complex-conjugation, conjugation by a unitary matrix, and their compositions. We include an overview of such symmetry maps on unital $C^\ast$-algebras. We say that $A\in\mathcal A$ is $\varphi$-symmetric if $\varphi(A)=A$, $A$ is $\varphi$-antisymmetric if $\varphi(A)=-A$, and $A$ has a $\zeta=e^{i\theta}$ $\varphi$-phase symmetry if $\varphi(A)=\zeta A$. Our main result is a new projection characterization of two operators $U$ (unitary), $B$ that have nearby commuting operators $U'$ (unitary), $B'$. This can be used to ``bootstrap'' symmetry from operators $U, B$ that are nearby some commuting operators $U', B'$ to prove the existence of nearby commuting operators $U'', B''$ which satisfy the same symmetries/antisymmetries/phase symmetries as $U, B$, provided that the symmetry maps and symmetries/antisymmetries/phase symmetries satisfy some mild conditions. We also prove a version of this for $X=U$ self-adjoint instead of unitary. As a consequence of the prior literature and the results of this paper, we prove Lin's theorem with symmetries: If a $\varphi$-symmetric matrix $A$ is almost normal ($\|[A^\ast, A]\|$ is small), then it is nearby a $\varphi$-symmetric normal matrix $A'$. We also extend this further to include rotational and dihedral symmetries. We also obtain bootstrap symmetry results for two and three almost commuting self-adjoint operators. As a corollary, we resolve a conjecture of arXiv:1502.03498 for two almost commuting self-adjoint matrices in the Atland-Zirnbauer symmetry classes related to topological insulators.