  We show that Connes' embedding conjecture (CEC) is equivalent to a real
version of the same (RCEC). Moreover, we show that RCEC is equivalent to a
real, purely algebraic statement concerning trace positive polynomials. This
purely algebraic reformulation of CEC had previously been given in both a real
and a complex version in a paper of the last two authors. The second author
discovered a gap in this earlier proof of the equivalence of CEC to the real
algebraic reformulation (the proof of the complex algebraic reformulation being
correct). In this note, we show that this gap can be filled with help of the
theory of real von Neumann algebras.
