Von Neumann entropy of the angle operator between a pair of intermediate subalgebras

TL;DR

This work analyzes the angle operator $\Theta = e_C e_D$ between two intermediate simple $C^*$-subalgebras in a unital inclusion $B\subset A$ with finite Watatani index and studies its Fourier dual $\mathcal{F}(\Theta)$ on relative commutants. Using the Fourier framework and auxiliary projections from Watatani-BG theory, the authors derive explicit formulas for the von Neumann entropy of $|\mathcal{F}(\Theta)|^2$ in the irreducible setting (Theorem A) and in the co-commuting square setting (Theorem B), namely $H(|\mathcal{F}(\Theta)|^2)=\dfrac{2}{\sqrt{[A:B]_0}}\eta(\delta\mathrm{tr}(e_Ce_D))$ with $\delta^2=[A:B]_0$, and $H(|\mathcal{F}(\Theta)|^2)=\dfrac{2}{\sqrt{[A:B]_0}}\eta\left( \dfrac{\sqrt{[A:B]_0}}{[A:C]_0 [A:D]_0} \right)$ respectively. They also establish a lower bound relating $H(|\Theta|^2)$ and $H(|\mathcal{F}(\Theta)|^2)$ and characterize the commuting square case by vanishing $H(|\Theta|^2)$ and a specific entropy for the dual; a converse result is provided when $H(|\Theta|^2)=0$. The results give computable entropy diagnostics for the relative position of $C$ and $D$ in $A$, and offer a concrete criterion to detect commuting vs. non-commuting configurations in this noncommutative setting.

Abstract

Given a pair of intermediate $C^*$-subalgebras of a unital inclusion of simple $C^*$-algebras with a conditional expectation of finite Watatani index, we discuss the corresponding angle operator and its Fourier transform. We provide a calculable formula for the von Neumann entropy of the (Fourier) dual angle operator for a large class of quadruple of simple $C^*$-algebras.

Theorems & Definitions (17)

• Theorem 2.1: BG, Theorem 3.5
• Lemma 2.2: BG, Lemma 4.2
• Proposition 2.3: BG, Lemma 4.4(2) and Proposition 4.6
• Definition 3.1
• Definition 3.2: JZW
• Proposition 3.3
• Proposition 3.4
• Proposition 3.5
• Lemma 3.6: BG, Proposition $5.15$
• Lemma 3.7