Michael Grosser

The norm-strict bidual of a Banach algebra and the dual of Cu(G)

To each Banach algebra A we associate a (generally) larger Banach algebra A+ which is a quotient of its bidual A″. It can be constructed using the strict topology on A and the Arens product on A″. A+ has certain more pleasant properties than A″, e.g. if A has a bounded right approximate identity, then A+ has a two-sided unit. In the special case A=L1(G) (G a locally compact abelian group) one gets A+=Cu(G)′, the dual of the space of bounded, uniformly continuous functions on G, and we show that the center of the convolution algebra Cu(G)′ is precisely the space M(G) of finite measures on G.

Algebra involutions on the bidual of a Banach algebra

Let A be a Banach algebra with bounded approximate right identity. We show that a necessary condition for the bidual of A to admit an algebra involution (with respect to the first Arens product) is that A*A=A*, i.e. the dual of A has to be essential as a right A-module. In particular, for any infinite, non-discrete, locally compact Hausdorff group G, L1(G)** does not admit any algebra involution with respect to either Arens product. This implies that the main result of a paper of R.S. Doran and W. Tiller concerning L1(G)** as Banach *-algebra (see [DT]) applies only to the trivial case of finite abelian groups.

Arens semi-regular Banach algebras

For some important Banach algebras, the first and the second Arens product on their biduals are different, i. e. these algebras are not (Arens) regular. Arens semi-regularity is a property strictly weaker than regularity; it characterizes those non-regular algebras (having a bounded two-sided approximate identity) for which the Arens products, though different, still behave in a reasonable way. The definition of semi-regularity is based on the relation of two natural embeddings of the space of double multipliers into the bidual of the Banach algebra. It is shown that each commutative Banach algebra is semi-regular and that semi-regularity is equivalent to the equality of the Arens products on certain subspaces of the bidual. Among others, group algebras and algebras of compact operators are treated as examples.

Ordinary Differential Equations in Algebras of Generalized Functions

A local existence and uniqueness theorem for ODEs in the special algebra of generalized functions is established, as well as versions including parameters and dependence on initial values in the generalized sense. Finally, a Frobenius theorem is proved. In all these results, composition of generalized functions is based on the notion of c-boundedness.

Colombeau’s Theory of Generalized Functions

The theory of distributions, founded by S. L. Sobolev and L. Schwartz, shows great power and flexibility in its natural domain, the theory of linear partial differential equations. Over the past five decades, numerous publications have contributed to an elaborate solution concept for such equations. As an example we mention the Malgrange-Ehrenpreis theorem showing that any constant coefficient linear PDE possesses a fundamental solution within the space of distributions. However, the inherent limitations of distribution theory, even within the realm of linear PDEs, became apparent as soon as 1957 when H. Lewy ([Lew57]) gave an example of a linear PDE with smooth coefficient functions without solutions in D′. Moreover, its structure as a space of linear functionals does not lend itself to a definition of a “multiplication” of distributions. In fact, various “impossibility results” show that an associative, commutative product on D′ would not coincide with various “natural” products on subspaces of D′. Nevertheless, there are quite a number of instances displaying a need for a concept of multiplication of distributions. Here is a list of some of them: 1. Nonlinear PDEs with singular data or coefficients (shock waves in systems from hydrodynamics and elasticity, delta waves in semilinear hyperbolic equations with rough initial data, propagation of acoustic waves in discontinuous media, Schrodinger equations with strongly singular potential, nonlinear stochastic PDEs with white noise excitation, Lie group transformations of generalized functions, ...). 2. Intrinsic problems in distribution theory (restriction to submanifolds, calculation of convolutions via Fourier transform, ...). 3. Renormalization problems in quantum field theory. 4. Singularities in nonlinear field theories, in particular in general relativity (ultrarelativistic limits of spacetime metrics, distributional curvature of cosmic strings, geodesic equations in distributional geometries ...). 5. Microlocal regularity and propagation of singularities in nonlinear PDEs or in linear PDEs with non-smooth coefficients.

Diffeomorphism Invariant Colombeau Theory

The main topic of this chapter is the presentation of the diffeomorphism invariant full Colombeau algebra G d (Ω) as it was given in [Gro01], yet with a considerable simplification concerning the definition of the ideal N. After envisaging sheaf-theoretic properties of G d (Ω) and some applications, we discuss a family of related algebras to highlight the general constraints for constructing diffeomorphism invariant full Colombeau algebras. Let us anticipate at this point that G d (Ω)—which can be considered as the “local” case—will be the basis for the construction of the intrinsically defined full Colombeau algebras on a general smooth manifold in Section 3.3.

Applications to General Relativity

The aim of this chapter is a detailed study of applicability and applications of distributional geometry as developed in the previous chapters—with its special focus, of course, on the generalized pseudo-Riemannian geometry of Section 3.2.5—to the theory of general relativity. We shall begin with a general introduction into our theme in Section 5.1, where we also introduce our notational conventions concerning general relativity. Section 5.2 is divided into two parts. The first one is concerned with a brief overview of applications of linear distributional geometry to relativity with a strong focus on a “no-go”-theorem by Geroch and Traschen (cf. [Ger87]). In the second part of Section 5.2 we use the nonlinear distributional geometry of Chapter 3 to define the generalized curvature tensor as well its contractions appearing in the field equations of general relativity. Moreover, we present a guideline for applying this setting to the study of singular spacetime metrics and discuss consistency with respect to the classical theory. Finally in Section 5.3 we give a complete distributional description of impulsive gravitational pp-waves based upon a variety of concepts introduced so far.

Applications to Lie Group Analysis of Differential Equations

In this chapter, as a first application of the concepts developed so far to geometric questions we develop an extension of Lie group analysis of differential equations to the generalized functions context. The methods we will use will enable us to treat the cases of distributional, Colombeau and weak (or integral) solutions simultaneously.

Generalized Functions on Manifolds

In this chapter we present a theory of generalized functions on manifolds as well as of generalized sections of vector bundles providing a framework for linear and nonlinear distributional geometry.