Angles Between Intermediate Operator Subalgebras

Motivated by [2] and [5], the notions of interior and exterior angles between a pair of compatible intermediate W*-subalgebras of an inclusion of W*-algebras with a normal conditional expectation with finite probabilistic index are introduced. This is then employed effectively to define the interior angle between a pair of compatible intermediate C*-subalgebras of an inclusion of non-unital C*-algebras with a conditional expectation with finite Watatani index. It is also shown that the interior angle is stable under the minimal tensor product of unital C*-algebras.

💡 Research Summary

The paper introduces and develops the concepts of interior and exterior angles between compatible intermediate subalgebras in the setting of operator algebras, extending earlier work that was limited to subfactors of II₁‑factors and to unital C*-algebras with finite Watatani index. The authors begin by reviewing the necessary background on Hilbert C*- and W*-modules, conditional expectations, and the various notions of index (probabilistic, completely positive, and Watatani). They emphasize the role of Hilbert W*-modules, especially the self‑dual modules introduced by Paschke, as the natural framework for handling normal conditional expectations with finite probabilistic index.

In the first substantive part, the paper treats an inclusion of von Neumann algebras (N\subset M) equipped with a normal conditional expectation (E_N:M\to N) of finite probabilistic index. For two compatible intermediate W*-subalgebras (P) and (Q) (compatibility meaning that the expectations restrict appropriately and the associated Jones projections commute), the authors define the interior angle (\theta(P,Q)=\arccos|e_P e_Q|) and the exterior angle (\phi(P,Q)=\arcsin|e_P e_Q|), where (e_P) and (e_Q) are the Jones projections acting on the standard Hilbert W*-module (L^2(M,E_N)). This definition generalizes the earlier subfactor angle and retains key properties such as symmetry, triangle inequality, and independence from the choice of quasi‑basis.

The second major section moves to non‑unital C*-algebras. Given an inclusion (B\subset A) with a faithful conditional expectation (E:A\to B) of finite Watatani index, the authors employ Izumi’s technique of passing to the enveloping von Neumann algebras (A^{}) and (B^{}). The normal extension (E^{}:A^{}\to B^{**}) inherits the finite probabilistic index, allowing the previously defined W*-angle to be applied to the lifted intermediate von Neumann subalgebras. By pulling back to the original C*-setting, they obtain a well‑defined interior angle between compatible intermediate C*-subalgebras of a non‑unital inclusion. The construction is shown to be independent of the particular quasi‑basis used in Watatani’s basic construction and to respect the reduced C*-basic construction (C^*_r\langle A,e_E\rangle).

A further contribution is the proof that the interior angle is stable under minimal tensor products of unital C*-algebras. For unital algebras (A_1, A_2) with intermediate subalgebras (P_i, Q_i) (i=1,2), the authors show that the Jones projections satisfy (|e_{P_1\otimes P_2}e_{Q_1\otimes Q_2}|=|e_{P_1}e_{Q_1}|\cdot|e_{P_2}e_{Q_2}|). Consequently, \