Conditional Expectations

In this section, we introduce a special kind of positive linear mapping, called a conditional expectation, and give some applications to tensor products.

Let be a C*-algebra and a C*-subalgebra

A linear mapping is called a projection if Φ(b) = b for every. In this case and.

A linear mapping is called -linear if Φ(ab) = Φ(a)b and Φ(ba) = bΦ(a) for every.

A -linear projection which is also a positive mapping, that is, is called a conditional expectation.

Theorem (J. Tomiyama). Every projection of norm 1 of the C*-algebra onto the C *-subalgebra is a conditional expectation.

Proof. Consider first a norm 1 projection of a W*-algebra onto a W*-subalgebra Then is a W*-algebra and its unit element is a projection.

Let e be a projection in, put f = 1 − e, and assume that either e or f is in. For any, both ex and fx are then in and for we have

So for

As is arbitrary, we have, that is, Interchanging e andf, we have. Thus

Putting Let be any positive form on and Since by ([L], 5.4.) it follows that 𝜑 is positive. Hence Φ is positive and so self-adjoint. Taking adjoints in (1) we have

Since the W*-algebra is the closed linear span of its projections ([L], 2.23), by (1) and (2) it follows that for

In the general case, when Φ is a projection of norm 1 of the C*-algebra onto its C*-subalgebra, we consider the second transpose, mapping of the second dual W*-algebra onto the second dual W*-algebra(A.15, A.16).

The first edition of this book appeared in 1981 as a direct continuation of Lectures of von Neumann Algebras by Ş.V. Strătilă and L. Zsidó and, until 2003, was the only comprehensive monograph on the subject.

The book Lectures on von Neumann Algebras, 2nd Edition, Cambridge University Press, 2019, will be always referred to as [L]. It is assumed that the reader is familiar with the material contained in this book, including the terminology and notation.

The present book contains the continuous decomposition and the discrete decomposition for factors of type III and all the necessary results such as the extensive theory of normal weights including the U. Haagerup characterization of normality, the A. Connes theorem of Radon–Nikodym type and the Pedersen–Takesaki construction, the conditional expectations and the operator-valued weights, a detailed consideration of groups of automorphisms and their spectral theory, and the theory of crossed products. In order to include our extension of the Takesaki duality theorem to noncommutative groups of automorphisms, we considered a simultaneous generalization of groups and duals of groups, namely the Kac algebras, and their actions on von Neumann algebras, as well as the corresponding crossed products. Instead of Kac algebras we can also consider quantum groups, the arguments being exactly the same as for Kac algebras.

In this second edition we added information and references for several results, which appeared after 1981, untill now, including A. Ocneanu's theorem concerning actions of amenable groups and its extensions, H. Kosaki's extension of the index to arbitrary factors and F. Rădulescu's examples of non-hyperfinite factors of type III, and of type III1 relying on D.-V. Voiculescu's theory of free probability.

For subjects such as the equivalence of injectivity and hyperfiniteness (originally due to A. Connes) we indicate references for more recent shorter proofs. This guarantees the uniqueness of the hyperfinite factor of type II∞ and of type factors,. Another important result which could not be considered in detail, due to its length, is the proof of U. Haagerup of the A. Connes conjecture that the hyperfinite factor of type III1 is unique.

Groups of Isometries on Banach Spaces

In this section, we describe the general framework for the spectral analysis of groups of isometrics on Banach spaces.

Let be a Banach space and a closed linear subspace. Besides the norm topologies, we shall also consider the weak topologies on and on. Consider the following conditions on the pair:

Lemma 1. Let be a pair-satisfying conditions and a bounded regular Borel measure on a separable locally compact Hausdorff space S. For every w-continuous norm-bounded function x there exists a unique element such that

Proof. The equation

defines a linear form on. To prove the lemma it is sufficient to show that is - continuous or, equivalently, that f is continuous with respect to the Mackey topology. Thus, we have to show that there exist an absolutely convex w-compact set and such that

We first assume that C = supp is compact. Then we have

Since C is compact and is w-continuous, the set is w-compact. Then the set is w-compact, and hence is absolutely convex and w-compact by condition (2). Since, inequality (2) follows from (3).

In the general case, there exists an increasing sequence of compact subsets of S such that. By the first part or the proof, there exist such that

Then, for every we have

and using condition it follows that is a Cauchy sequence in. If is the limit of this sequence, it then follows from (4) that for all .

The uniqueness of the element satisfying (1) follows obviously using (1x).

The unique element satisfying (1) will be denoted by

Consider now two pairs and satisfying conditions and. Let be the Banach space or all bounded linear operators and the linear space or all w-continuous linear operators. Using the Banach–Steinhauss theorem, it is easy to check that is a norm-closed linear subspace. In particular, is a Banach space.

Characterizations of Normality

In this section, we prove the Theorem of Haagerup asserting that every normal weight on a W *-algebra is the pointwise least upper bound of the normal positive forms it majorizes.

Let be a C*-algebra. A weight on is a mapping with the properties

The set

is a face of, the set

is a left ideal of, and the set

is a facial subalgebra of, hence can be extended uniquely to a positive linear form, still denoted by, on the *-algebra

A family of weights on is called sufficient if

and is called separating if

In particular, the weight is called faithful if

Let be a weight on the C*-algebra. The formula

defines a prescalar product on with the properties:

Let be the Hilbert space associated with with the scalar product It follows that there exists a *-representation, uniquely determined, such that

where denotes the canonical mapping. The *-representation is called the GNS representation or the standard representation associated with We remark that

Let be a W *-algebra. A weight on is called normal if

for every norm-bounded increasing net, and lower w-semicontinuous if the convex sets

are w-closed. An important result concerning weights on W *-algebras is the following characterization of normality:

Theorem (U. Haagerup). Let 𝜑 be a weight on the W *-algebra. The following statements are equivalent:

(i) is normal;

(ii) is lower w-semicontinuous;

Later (2.10, 5.8) we shall see that is normal if and only if it is a sum of normal positive forms, in accordance with the definition used in ([L], 10.14).

In Sections 1.4–1.7, we present some general results that will be used in the proof of the theorem; Sections 1.6–1.12 contain the main steps of the proof.