Unitary orthonormal bases of finite dimensional inclusions
Keshab Chandra Bakshi, B V Rajarama Bhat
TL;DR
This work extends the theory of unitary orthonormal bases for finite dimensional inclusions to general subalgebra systems $(\mathcal{B}\subseteq \mathcal{A},E)$. It establishes necessary spectral and trace conditions, notably $A^t\tilde{n}=d\tilde{m}$ with $\tilde{m},\tilde{n}$ PF-eigenvectors and $E$ preserving the Markov trace, and then constructs explicit UOBs in wide cases, including abelian and Weyl-like generalizations. The paper shows stability under tensor products, direct sums, and basic construction, yielding broad classes of inclusions with UOBs. Two main applications follow: (i) depth-2 subfactors with abelian relative commutants and superextremal inclusions possess UOBs, implying integer indices, and (ii) the Connes–Størmer relative entropy for the basic construction satisfies $H(\mathcal{A}_1|\mathcal{A})=\ln \|A\|^2$ when a $\tau$-preserving E has a UOB. These results unify and generalize CKP and Weyl-unitary constructions and illuminate connections to subfactor invariants and quantum information structures through explicit unitary bases.
Abstract
We study unitary orthonormal bases in the sense of Pimsner and Popa for inclusions $(\mathcal{B}\subseteq \mathcal{A}, E),$ where $\mathcal{A}, \mathcal{B}$ are finite dimensional von Neumann algebras and $E$ is a conditional expectation map from $\mathcal{A}$ onto $\mathcal{B}$. It is shown that existence of such bases requires that the associated inclusion matrix satisfies a spectral condition forcing dimension vectors to be Perron-Frobenius eigenvectors and the conditional expectation map preserves the Markov trace. Subject to these conditions, explicit unitary orthonormal bases are constructed if either one of the algebras is abelian or simple. They generalize complex Hadamard matrices, Weyl unitary bases, and a recent work of Crann et al which correspond to the special cases of $\mathcal{A}$ being abelian, simple, and general multi-matrix algebras respectively with $\mathcal{B}$ being the algebra of complex numbers. For the first time $\mathcal{B}$ is more general. As an application of these results it is shown that if $(\mathcal{B}\subseteq \mathcal{A}, E),$ admits a unitary orthonormal basis then the Connes-Størmer relative entropy $H(\mathcal{A}_1|\mathcal{A})$ equals the logarithm of the square of the norm of the inclusion matrix, where $\mathcal{A}_1$ denotes the Jones basic construction of the inclusion. As a further application, we prove the existence of unitary orthonormal bases for a large class of depth 2 subfactors with abelian relative commutant.