Remarks on free mutual information and orbital free entropy

Abstract

The present notes provide a proof of $i^{\ast }(\mathbb{C}P+\mathbb{C}(I-P);\mathbb{C}Q+\mathbb{C}(I-Q\left)\right)=-{\chi }_{\mathrm{orb}}(P,Q)$ for any pair of projections $P,Q$ with $\tau \left(P\right)=\tau \left(Q\right)=1/2$. The proof includes new extra observations, such as a subordination result in terms of Loewner equations. A study of the general case is also given.