We show that Connesʼ 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.

Additional Metadata
Keywords Connesʼ embedding problem, Real von Neumann algebras
MSC General theory of von Neumann algebras (msc 46L10)
THEME Other (theme 6)
Publisher Academic Press
Journal Advances in Mathematics
Burgdorf, S, Dykema, K, Klep, I, & Schweighofer, M. (2014). Addendum to “Connesʼ embedding conjecture and sums of hermitian squares” [Adv. Math. 217 (4) (2008) 1816–1837]. Advances in Mathematics, 252, 805–811.