On conditional expectations of finite index

TL;DR

This paper characterizes when a conditional expectation on a unital C*-algebra has finite index, linking positivity and complete positivity, and explores the existence of Jones' towers and their properties in various algebraic settings.

Contribution

It provides a new characterization of finite index conditional expectations, extending previous results and describing their behavior in the context of W*- and C*-algebras.

Findings

The existence of a finite index is characterized by positivity conditions.

Jones' towers always exist in the W*-case and under certain C*-conditions.

Finite index expectations commute with specific projections in the algebra centers.

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.