On conditional expectations of finite index
M. Frank, E. Kirchberg
TL;DR
This work extends the notion of conditional expectations to finite index in the C*-algebra setting by establishing equivalences between faithfulness with a positive map bound $K$ and complete positivity bounds $L$, via $K(E)$ and $L(E)$. It defines the index ${\rm Ind}(E)$ for finite-index conditional expectations as the discrete-part projection of ${\rm Ind}(E^{**})$, and analyzes when the Jones tower can be constructed, proving existence in the W*-case and in the C*-case when ${\rm Ind}(E)\in {\rm Z}(B)$. The paper shows that normal conditional expectations of finite index commute with central projections and canonical type decompositions, yielding decomposition-preserving properties and structural consequences for relative commutants and dimensions, with non-commutative-topology interpretations and dimension-estimation formulas. These results connect the operator-algebraic index theory of Pimsner–Popa, Baillet–Denizeau–Havet, Watatani, and Popa to a broader C*-framework, clarifying when a finite-index tower exists and how index data localizes to discrete components, including notable examples outside the base algebra and implications for non-separable and type III settings.
Abstract
For a conditional expectation E on a (unital) C*-algebra A there exists a real number K>=1 such that the mapping (K.E-id_A) is positive if and only if there exists a real number L>=1 such that the mapping (L.E-id_A) 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; M. Baillet, Y. Denizeau, J.-F. Havet; Y. Watatani, 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) in E(A) in the C*-case. Furthermore, normal conditional expectations of finite index commute with the general W*-projections to their finite, infinite, discrete and continuous type I, type II_1, type II_\infty and type III parts, i.e. the respective projections in the centers of the initial and the image W*-algebra coincide. We give an interpretation of our result in terms of non-commutative topology and indicate some dimension estimation formulae and an inequality.