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We exhibit an operator norm bounded, infinite sequence { A n } \{A_n\} of 3 n × 3 n 3n \times 3n complex matrices for which the commutator map X ↦ X A n − A n X X\mapsto XA_n - A_nX is uniformly bounded below as an operator over the space of trace-zero self-adjoint matrices equipped with Hilbert–Schmidt norm. The construction is based on families of quantum expanders. We give several potential applications of these matrices to the study of quantum expanders. We formulate several natural conjectures and provide numerical evidence. 

Abstract This paper gives a free entropy theoretic perspective on amenable absorption results for free products of tracial von Neumann algebras. In particular, we give the 1st free entropy proof of Popa’s famous result that the generator MASA in a free group factor is maximal amenable, and we partially recover Houdayer’s results on amenable absorption and Gamma stability. Moreover, we give a unified approach to all these results using $$1$$-bounded entropy. We show that if $${\mathcal{M}} = {\mathcal{P}} * {\mathcal{Q}}$$, then $${\mathcal{P}}$$ absorbs any subalgebra of $${\mathcal{M}}$$ that intersects it diffusely and that has $$1$$-bounded entropy zero (which includes amenable and property Gamma algebras as well as many others). In fact, for a subalgebra $${\mathcal{P}} \leq{\mathcal{M}}$$ to have this absorption property, it suffices for $${\mathcal{M}}$$ to admit random matrix models that have exponential concentration of measure and that “simulate” the conditional expectation onto $${\mathcal{P}}$$.