6 O ct 1 99 8 On conditional expectations of finite index
نویسنده
چکیده
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 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. 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.
منابع مشابه
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 con...
متن کامل1 5 A pr 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 entire part of a real number. As a cons...
متن کاملar X iv : c ha o - dy n / 99 10 00 5 v 1 6 O ct 1 99 9 The Finite - Difference Analysis and Time Flow
متن کامل
Lecture Notes on Probability
Disclaimer 3 1. Why we need measure theory 4 1.1. Riddle 1 4 1.2. Riddle 2 4 1.3. Riddle 3 4 1.4. Why we need measure theory 4 2. Measure theory 6 2.1. π-systems, algebras and sigma-algebras 6 3. Carathéodory’s Theorem and constructing measures 8 4. Events and random variables 10 5. Independence 13 6. The tail sigma-algebra 16 7. Expectations 19 8. A strong law of large numbers and the Chernoff...
متن کامل1 6 Ju n 20 04 Intersections of finite families of finite index
We prove that finiteness of the index of the intersection of a finite set of finite index subalgebras in a von Neumann algebra (with small centre) is equivalent to the finite dimensionality of the algebra generated by the conditional expectations onto the subalgebras. Supported in part by NSF gramt DMS-9322675 and Marsden grant UOA520. Supported in part by NSF grant DMS-0200770.
متن کامل