Remarks on free mutual information and orbital free entropy

Abstract

The present notes provide a proof of $i^{\ast }(\mathbb{C}P+\mathbb{C}(I-P);\mathbb{C}Q+\mathbb{C}(I-Q))=-{\chi }_{\mathrm{orb}}(P,Q)$ for any pair of projections $P,Q$ with $\tau (P)=\tau (Q)=1/2$. 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.

13 pages

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