Abstract
We show that the spectral measure of any non-constant non-commutative polynomial evaluated at a non-commutative n-tuple cannot have atoms if the free entropy dimension of that n-tuple is n (see also work of Mai, Speicher, and Weber). Under stronger assumptions on the n-tuple, we prove that the spectral measure of any non-constant non-commutative polynomial function is not singular, and measures of intervals surrounding any point may not decay slower than polynomially as a function of the interval's length.
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