# Journal of Operator Theory

Volume 40, Issue 1, Summer 1998 pp. 87-111.

On conditional expectations of finite index**Authors**: Michael Frank (1), and Eberhard Kirchberg (2)

**Author institution:**(1) Universit\"at Leipzig, FB Mathematik/Informatik, Mathematisches Institut, Augustusplatz 10, D-04109 Leipzig, Germany

(2) Humboldt-Universit\"at zu Berlin, FB Mathematik, Institut f\"ur reine Mathematik, Ziegelstr.~13 A, D-10117 Berlin, Germany

**Summary:**For a conditional expectation $E$ on a (unital) $C^*$-algebra $A$ there exists a real number $K \geq 1$ such that the mapping $K \cdot E - {\rm id}_A$ is positive if and only if there exists a real number $L \geq 1$ such that the mapping $L \cdot E - {\rm id}_A$ is completely positive, among other equivalent conditions. The estimate $(\min K) \leq (\min L) \leq (\min K) [\min K]$ is valid, where $[\cdot]$ denotes the entire part of a real number. As a consequence the notion of a ``conditional expectation of finite index'' is identified with that class of conditional expectations, which extends and completes results of M.~Pimsner, S.~Popa ([27], [28]), M.~Baillet, Y.~Denizeau and J.-F.~Havet ([6]) and Y.~Watatani ([35]) and others.

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