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Let $N\subset M$ be a finite index $II_1$-subfactor. Let $B=\{b_i\}$ be a finite orthonormal (Pimsner-Popa) basis for $M$ over $N$. Let $d=[M\colon N]^{1/2}$. It is well known that $B_1=\{d b_{i_1} e_1 b_{i_2}\}$ is a (not necessarily orthonormal) basis for $M_1$ over $N$, where $M_1=\langle M, e_1\rangle$ is the basic construction of $N\subset M$ (for example, see Bisch, Bimodules and higher relative commutants, page 31). Under what conditions are we assured $B_1$ is orthonormal? orthogonal?

Integer index always works (see the Pimsner-Popa paper), and a variation always works in infinite index (with a definition of orthonormal basis due to Burns).

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