For positive-semidefinite matrices $A, B$ in $M_n(\mathbb{C})$, the Minkowski determinant theorem tells us that $\det(A+B)^{\frac{1}{n}} \ge \det(A)^{\frac{1}{n}} + \det(B)^{\frac{1}{n}}$. For a matrix $A$ in $M_n(\mathbb{C})$, the Fuglede-Kadison determinant is given by $\Delta(A) = |\det(A)|^{\frac{1}{n}}$. A natural guess is that an inequality of the form $\Delta(A+B) \ge \Delta(A) + \Delta(B)$ should hold for positive operators $A, B$ in a $II_1$-factor. I believe I have seen this result in the literature a while ago (I could be misremembering). But now when I search for it, it seems to completely elude me. It would be of great help if someone could direct me to a reference with the original version of the proof. Thank you.
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$\begingroup$ Being a new member, as I cannot add comments to the original post, I am posting it as an answer. A quick proof would involve the convexity of $\ln(1+e^x)$ and Jensen's inequality for the faithful normal tracial state of the $II_1$-factor. Just as a clarification, I am as much interested in references and mentions of this result in the literature, as (other) proofs. $\endgroup$– nsoumCommented Mar 8, 2017 at 15:16
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