Deformed Joint Free Distributions of Semicircular Elements Induced by Multi Orthogonal Projections

Abstract

In this paper, we consider (i) how to establish semicircular elements \(\{U_{k}\}_{k=1}^{N}\) induced by N-many mutually orthogonal projections \(\{q_{k}\}_{k=1}^{N}\), for N \(\in \) \((\mathbb {N}\setminus \{1\})\cup \{\infty \},\) and the corresponding free product Banach \(*\)-probability space \(\mathbb {L}_{Q}^{(N)}\) generated by \(\{U_{k}\}_{k=1}^{N}\), (ii) the free-distributional data on \(\mathbb {L}_{Q}^{(N)}\), (iii) certain \(*\)-homomorphisms on \(\mathbb {L}_{Q}^{(N)},\) and (iv) how the \(*\)-homomophisms of (iii) deform the original free-distributional data of (ii).