Remarks on free mutual information and orbital free entropy

英文摘要：

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