8 O ct 1 99 8 On conditional expectations of finite index

For a conditional expectation E on a (unital) C*-algebra A there exists a real number K ≥ 1 such that the mapping K · E − idA is positive if and only if there exists a real number L ≥ 1 such that the mapping L ·E − idA is completely positive, among other equivalent conditions. The estimate (min K) ≤ (min L) ≤ (min K)[min K] is valid, where [·] denotes the integer 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. Situations for which the index value and the Jones’ tower exist are described in the general setting. In particular, the Jones’ tower always exists in the W*-case and for Ind(E) ∈ E(A) in the C*-case. Furthermore, normal conditional expectations of finite index commute with the (abstract) projections of W*-algebras to their finite, infinite, discrete and continuous type I, type II1, type II∞ and type III parts, i.e. they respect and preserve these W*-decompositions in full.