For inclusions of II₁ factors and finite-dimensional algebras, the logarithmic Pimsner-Popa index equals the supremum, over all states and all Rényi parameters p in [1/2,∞], of the sandwiched Rényi relative entropy to the subalgebra.
Relative entropy of states of von neumann algebr as
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Relative entropy for von Neumann subalgebras
For inclusions of II₁ factors and finite-dimensional algebras, the logarithmic Pimsner-Popa index equals the supremum, over all states and all Rényi parameters p in [1/2,∞], of the sandwiched Rényi relative entropy to the subalgebra.