We show that in a tracial and finitely generated
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815301183&_mathId=si1.gif&_user=111111111&_pii=S0001870815301183&_rdoc=1&_issn=00018708&md5=def7dfa470e17ec3403bb5a2f56f5c72" title="Click to view the MathML source">W⁎class="mathContainer hidden">class="mathCode">-probability space existence of conjugate variables excludes algebraic relations for
the generators. Moreover, under
the assumption of maximal non-microstates free entropy dimension, we prove that
there are no zero divisors in
the sense that
the product of any non-commutative polynomial in
the generators with any element from
the von Neumann algebra is zero if and only if at least one of those factors is zero. In particular, this shows that in this case
the distribution of any non-constant self-adjoint non-commutative polynomial in
the generators does not have atoms.
Questions on the absence of atoms for polynomials in non-commuting random variables (or for polynomials in random matrices) have been an open problem for quite a while. We solve this general problem by showing that maximality of free entropy dimension excludes atoms.