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.

The discovery of the modular operator and the modular automorphism group associated with a normal semifinite faithful weight has led to a powerful theory – the modular theory – which is nowadays essential to the consideration of many problems concerning operator algebras. This theory has been developed in close association with the effort to understand the structure and produce examples and refined classifications of factors. Thus, the crossed product construction, which gave rise to the first non-trivial examples of factors, has been shown to play a fundamental role in the structure theory as well, by reducing the study of the purely infinite algebras to the study of the more familiar semifinite algebras and their automorphisms. Moreover, several algebraic invariants, previously defined only in some special cases, have been introduced via the modular theory for arbitrary factors and the corresponding classification has been proved to be almost complete for approximately finite dimensional factors.

The present book is a unified exposition of the technical tools of the modular theory and of its applications to the structure and classification of factors. It is based on several works recently published in periodicals or just circulated as preprints. The main sources used in writing this book are the works of W. B. Arveson, A. Connes, U. Haagerup, M. Landstad, G. K. Pedersen, M. Takesaki, and J. Tomiyama. The general treatment of crossed products follows an article by Ş. Strătilă, D. Voiculescu and L. Zsidó. Due to the wealth and variety of results recently obtained, it has not been possible to include here a detailed exposition of the classification of injective factors and their automorphisms; these topics and several others are just mentioned in the Notes sections, together with appropriate references.

The reader is assumed to have a good knowledge of the general theory of von Neumann algebras, including the standard forms. Actually, the present book can be viewed as a sequel to a previous book, Ş. Strătilă and L. Zsidó – Lectures on von Neumann algebras, Editura Academiei & Abacus Press, 1979, which is often quoted here and referred to as [L]. There is also an Appendix which contains some supplementary results on positive self-adjoint operators and introduces the terminology connected with W*-algebras.

Hopf–von Neumann Algebras

In this section, we consider a category of objects called Hopf–von Neumann algebras, which in a certain sense generalize locally compact groups, and also their actions on W*-algebras. The main interest of these objects consists in their giving a natural framework for the duality theory of locally compact groups.

We first introduce certain notation and conventions that will be used frequently in what follows.

Let be Hilbert spaces and von Neumann algebras. There exists a unique unitary operator

such that for all. The mapping

is then a *-isomorphism, uniquely determined, such that

The value of a linear form on a vector will also be denoted by

Let G be a locally compact group. The elements of the Hilbert space will be identified with vector-valued functions on G. Also, the linear operators on will usually be defined by specifying the elements. The formal versions of these procedures are standard and well known.

The identity mapping on will be denoted by, or by, where indicates a position in tensor products. The same convention will be used for the unit element

A Hopf–von Neumann algebra is a pair, where is a W *-algebra and is an injective unital normal *-homomorphism, called comultiplication, which is coassociative, that is

The Hopf–von Neumann algebra is said to be commutative if the algebra is commutative and is called cocommutative if

A coinvolutive Hopf–von Neumann algebra is a triple, where is a Hopf–von Neumann algebra and is an involutive *-antiautomorphism, called coinvolution, such that

An action of the Hopf–von Neumann algebra on the W *-algebra is an injective unital normal *-homomorphism such that

In this case, we also say that is an -comodule via.

Note that is an action of on.

The centralizer of the action

Clearly, is a unital W *-subalgebra of.

Let be -comodules via the actions respectively. We shall say that a normal completely positive linear mapping intertwines the actions or that is an -comodule mapping if

In particular, if there exists a *-isomorphism intertwining then we say that the actions are isomorphic.

In this section, we review some facts used in the main text concerning positive self-adjoint operators in Hilbert spaces, W-algebras and infinite tensor products. Throughout this section, will denote a complex Hilbert space.

Let A be a positive self-adjoint linear operator on with domain. For each, we put (see [L], 9.9) and for each, we define ; note that. From ([L], 9.9–9.11) we see, using Lebesgue's dominated convergence theorem, that a vector belongs to if and only if

if, then we put, so that the above limit gives for any vector

Recall ([L], 9.20) that the restriction of A to is the analytic generator of the so-continuous unitary representation; the operators can be also regarded as partial isometries acting on the whole of.

Let a be a subset of. A continuous function is said to be operator continuous if the mapping

is continuous with respect to the so-topology and the s-topology.

Theorem (I. Kaplansky, R. V. Kadison). Let be such that and let be a continuous function such that. Then f is operator continuous.

Proof. We first prove the theorem for. Note that the functions and are operator continuous on. Indeed, if are normal operators such that and, then

A similar argument holds for the s-topology.

Denote by the set of all operator continuous functions on and by the subset of bounded functions in. Then is a uniformly closed self-adjoint vector subspace of the *-algebra of all continuous complex functions on and. Hence is a uniformly closed *-subalgebra containing the functions and. By the Stone–Weierstrass theorem, it follows that contains the *-subalgebra of continuous complex functions vanishing at infinity on. Let be a continuous function such that sup. Then belongs to and since it follows that belongs to (actually to), so that belongs to and we conclude that f, as a sum of two functions in, also belongs to.

Consider now the general case. Let be normal operators with Sp(xi) and Sp(x) contained in and. Since, there exists a compact neighborhood of such that. Then, hence is closed in and, by the Tietze– Urysohn extension theorem, f can be extended to a continuous function g on. Let be a continuous function such that supp and.

The Connes Invariant T(ℳ)

In this section, we introduce the invariant as the group of inner periods of modular automorphism groups on.

Let be a W*-algebra and the canonical quotient mapping. Recall (3.2) that the modular homomorphism defined by does not depend on the choice of the n.s.f. weight on and that is contained in the center of the group Out. Therefore, the kernel of the modular homomorphism is a subgroup of the additive group and an algebraic invariant of, which we denote by

Theorem. Let ℳ be a W*-algebra and. The following statements are equivalent:

(i) t ∈ T (ℳ);

(ii) for every we have Int;

(iii) for every there exists a unitary element such that;

(iv) for everythere exists a (strictly semifinite) commuting with such that

(v) there exists such that.

If ℳ is countably decomposable and, then

(vi) there exists a faithful normal state on with.

Proof. It is clear that (vi) ⇒ (iv) ⇒ (v) ⇒ (i) ⇒ (ii).

(ii)⇒(iii). If with, then for, whence. For, we have, so.

(iii) ⇒ (iv). By assumption, we have for some unitary. There exists a nonsingular positive self-adjoint operator A affiliated to such that. Then the n.s.f. weight commutes with and. On the other hand, we recall (10.9) that if, then the weight is strictly semifinite.

(iii) ⇒ (vi). Assume that is countably decomposable. By (iii) there exists an n.ss.f. weight on such that. Since is strictly semifinite and is countably decomposable, it follows by Theorem 10.9 that there exists a sequence with supports, mutually orthogonal, such that faithful normal state with.

Proposition. Let ℳ be a W*-algebra. If is semifinite, then. If and has separable predual, then is semifinite.

Proof. If is an n.s.f. trace on,for all, and. Conversely, if and is any n.s.f. weight on, then for all. If has separable predual, then, by Theorem 15.16, there exists an s-continuous unitary representation such that. Also, there exists a nonsingular positive self-adjoint operator A affiliated to such that and hence, for all, that is, is an n.s.f. trace on.

Thus, for any nonsemifinite W*-algebra ℳ with separable predual the center of the group Out(ℳ) is nontrivial.

Dominant Weights and Continuous Decompositions

Let be a properly infinite W*-algebra and denote by (resp.); resp. the set of all normal (resp. normal semifinite; resp. n.s.f.) weights on. In Sections 23.1–23.3, we consider a fixed weight and, using the canonical bijection given by Theorems 3.1 and 5.1, we apply the results obtained in Section 20 in order to get a comparison theory for weights. In particular, we define the notion of a dominant weight on, this yields the continuous decomposition of, the properties of which are analyzed further. The main feature of the continuous decomposition is that it reduces the study of type III W*-algebras to the study of actions of on semifinite W*-algebras, modulo (outer) conjugation.

Recall (2.21) that if and is a partial isometry with, then the Formula defines a weight with; moreover, if, then. Consider a fixed weight.

Proposition. Let and with and. Consider the W*-algebra and the balanced weight. The following statements are equivalent:

Proof. (i) ⇔ (ii). We have, and it also follows that; it also follows that. By Proposition

2.21, we have if and only. For, we have and, hence and only if.

(ii) ⇔ (iii). Consider now the W*-algebra and the weight defined as. Then by Proposition 3.3, we have

so that

hence if and only if for every.

If, then we have, and, so that the equivalence (ii) ⇔ (iii) also follows from Proposition 20.2.

By the previous proposition we have that for the following conditions are equivalent:

(i) there exists

(ii) there exists;

If these conditions are satisfied, we write and we say that the weight is dominated by the weigh. If in condition (i) we have also or, equivalently, in condition (iii) we have then we write and say that the weights and are equivalent. For the relations and we can thus apply all the results obtained in Section 20. In particular, If and, then.

Proposition. Let and be partial isometries such that. Then if and only if in.

Proof. Put. We have, and hence and, similarly,. Thus, we have to prove that if and only if in.

If in, there exists with and; hence, by Proposition 2.21.

Conversely, if then there exists with and.

This fourth chapter introduces the second canonical model of low Reynolds number swimming, namely that of the squirming sphere. Originally proposed by Lighthill (1952), and later extended by Blake (1971), this model is a variation of the waving sheet adapted to a finite-size swimmer. Here, the surface of a spherical body undergoes periodic small-amplitude deformations, leading to instantaneous velocity boundary conditions applied on an effective spherical frame. Since Lighthill's original paper, the squirmer model has been used and extended in a variety of setups, in particular to provide an alternative envelope model of the metachronal waves of cilia for finite-size organisms. We first derive the classical swimming squirmer model for a translating swimmer before discussing its extension to rotational motion. We then show how to link the motion of points on the surface of a deformable sphere described in a Lagrangian fashion to the squirmer model written naturally in an Eulerian framework. We finish by comparing the results of the model with flow measurements around the flagellated green alga Volvox.

In this fourteenth chapter we turn to the collective dynamics of cell populations such as bacterial colonies. Specifically, we use the results from Chapters 9 and 10 on hydrodynamic interactions to adapt the discrete and continuum frameworks introduced in the previous chapter to the case of collective cell locomotion. Swimming cells create flows, which advect and rotate neighbouring organisms, and since the flow induced by each cell depends on its location and orientation, this coupling leads to complex nonlinear swimming dynamics. In the discrete case, we derive a first-principle model of cells interacting in the dilute limit, demonstrate the different ways in which two swimming microorganisms affect each other hydrodynamically, and show how the model can be used to explain clustering instabilities of swimming algae. We then develop a continuum approach coupling the dynamics of the fluid with the distribution in position and orientation of the cell population. After relating the model to alternative phenomenological descriptions based on symmetry arguments, we use this continuum framework to capture collective cell instabilities.

In many biological situations, self-propelled cells and slender appendages interact with complex, non-Newtonian fluids. In this fifteenth and final chapter, we revisit the Newtonian hydrodynamic principles from Chapter 2 for fluids characterised by a nonlinear, or non-instantaneous, relationship between stress and deformation. We first consider linear viscoelastic fluids. We show how the presence of memory in the fluid affects the hydrodynamic forces and the energy of slender filaments, and compare a modified active-filament model of spermatozoa to experiments in viscoelastic fluids. We next address fluids with nonlinear rheological properties and their impact on the locomotion of Taylor's waving sheet. When the wave kinematics are prescribed, the nonlinear fluid leads to decreases in both speed and energy expenditure, which compares favourably to experiments using small nematodes. If instead the wave results from a balance between activity, elasticity and fluid forces, a transition to enhanced motion is possible for sufficiently flexible swimmers. We finish by addressing heterogeneous fluids and show how the multiscale nature of the fluid systematically enhances flow and transport.

In this first chapter, we give a brief biological introduction to understand the context of the mathematical models developed in the following chapters of the book and to also appreciate the relevance of the biophysical problems addressed. The chapter includes also a short overview of the role played by fluid dynamics in biology.