Introduction

In this chapter we investigate the role which minimal left ideals play in those algebras which possess them. (Minimal here simply means minimal under inclusion among all non-zero left ideals.) Significant results usually depend on assuming that the algebra is semiprime. However, the stronger assumption of semisimplicity, which is common in other areas of our theory, is comparatively seldom needed here. Furthermore, norms behave particularly well on minimal ideals so that a number of results can be proved for normed algebras without assuming completeness. In particular, any norm is spectral on many of the algebras considered in this chapter.

We will naturally be concerned mainly with algebras which not only have some minimal left ideals, but “enough” minimal left ideals in some sense. Such algebras axe quite special, but they have a correspondingly rich theory. Many classes of algebras with “enough” minimal left ideals have been defined, and towards the end of the section we will prove all the inclusions which hold between these various classes. (These results are summarized in a diagram of implications in Theorem 8.8.11.) However, we will concentrate attention on the class of modular annihilator algebras which was introduced by Bertram Yood [1958] and has been intensively studied by Yood [1964] and by Bruce A. Barnes [1964], [1966], [1968a], [1968b], and [1971a]. This class seems to give a particularly good compromise between axioms strong enough to give a significant theory and axioms weak enough to be satisfied by most important examples. Furthermore, modular annihilator algebras are defined purely algebraically while many of the other classes axe defined in terms of mixed algebraic and topological criteria. This provides a particularly elegant and straightforward theory.

This volume provides a gentle introduction to most of the main areas of research on general Banach algebras. It also serves the more specific purpose of providing the background for Volume II which will deal more intensively with *-algebras (i.e. algebras with fixed involutions, normally denoted by *). The focus is on the algebraic, and sometimes the geometric, underpinnings of the analytic theory. The subject is rich with aesthetic appeal, and many topics are pursued just as far as I found them attractive. References are given to more thorough expositions when they are available or to original sources. I have tried to make the book readable for beginning graduate students. Towards this end, I sometimes include a bit of undergraduate level material when it may not have been absorbed by such readers. There are also generous comments and historical remarks. They are all intended to serve a pedagogic purpose. I have tried to document the original source of most ideas, but sometimes I have failed. I apologize to those thus slighted. The knowledgeable reader will also find numerous previously unpublished results and technical improvements.

Readers should note the Symbol Index at the end of the volume. I have chosen notation carefully and used it consistently throughout the work. For instance, A always represents an algebra and A with a subscript always represents a subset of that algebra. Each entry in the bibliography displays the numbers of the sections in this volume to which it is related.

Introduction

This chapter begins by stating some basic conventions, definitions and notation that will be used throughout the work. Additional standard notations will be introduced from time to time, as needed. The reader should consult the index of notation for reference. Many of the ideas presented in the first section will be familiar to some readers. They are mentioned for the sake of review and to fix our notation. Also, of course, some standard concepts are defined in slightly different ways by different authors, and we wish to make clear our own conventions. The chapter concludes with a number of examples discussed in some depth. We urge readers to acquaint themselves with these since an abstract theory, such as that presented in this work, lacks substance without knowledge of examples.

The first section deals primarily with basic elementary results on normed, semi-normed, or topological linear spaces and algebras. Such topics as ideals, homomorphisms, quotient norms, etc. are discussed, and the role of semi-norms in locally convex topological linear spaces is quickly surveyed. The unitization of an algebra and an important convention about it are also introduced.

Let A be an algebra. In this chapter we will study the space 〈 PA / ΠA / ΞA 〉 of 〈 prime / primitive / maximal modular 〉 ideals of A as a topological space under the hull–kernel or Jacobson topology. For certain classes of algebras A, e.g., completely regular algebras (Section 7.2) and strongly harmonic algebras (Section 7.4), we will show that the subdirect product representation relative to ΞA (introduced in Definition 1.3.3 and Section 4.6) yields significant information about A. Section 7.3 deals with more detailed questions in ideal theory revolving around primary ideals. We also consider central and weakly central algebras and show that they are completely regular under fairly weak additional hypotheses.

The Hull-Kernel Topology

In Section 3.2 we introduced the hull–kernel topology on the Gelfand space ΓA of a commutative Banach algebra A. It is comparatively little used except in the case of completely regular commutative spectral algebras where it is Hausdorff and coincides with the Gelfand topology. In the commutative case, Proposition 3.1.3 shows that the Gelfand space of A can be identified with the set ΞA of maximal modular ideals of A, and Theorem 4.1.9 shows that the latter set coincides with the set ΠA of primitive ideals.

In a noncommutative algebra A, the set PA of prime ideals and its subsets, ΠA and ΞA, can each be given the hull–kernel topology. Again, this topology seems to be of comparatively little use unless further restrictions are placed on the algebra.

Introduction

In any mathematical theory it is important to compare the general abstract structures which arise with comparatively well understood concrete examples. Representation theory is a systematic attempt to do this. We have already seen one example of a representation theory in the Gelfand theory for commutative spectral algebras given in Chapter 3. Any commutative spectral algebra modulo its Gelfand radical is isomorphic to a spectral subalgebra of the algebra of all continuous functions vanishing at infinity on a locally compact Hausdorff space. Since algebras of continuous functions are comparatively concrete and can perhaps be somewhat better understood a priori than general commutative spectral algebras, this is an important (and characteristic) representation theory. Note that the class of all commutative spectral algebras is too large to be successfully represented. Only the semisimple commutative spectral algebras can be faithfully represented. Fortunately the pathology of the non-semisimple algebras can be neatly excised by dividing out the Gelfand radical.

In this work we will study several other representation theories in considerable detail. This chapter deals with the representation of general algebras and part of the second volume is devoted to the representation theory of *-algebras. In both these cases the phenomena noted above occur. Not all of the objects (algebras or *-algebras) can be faithfully represented, but the pathological part can be divided out.

Introduction

Multisummability of formal solutions of meromorphic differential equations was proved by J.-P.Ramis [9], J.Martinet and J.-P.Ramis [8], B.Malgrange and J.-P.Ramis [7], B.L.J.Braaksma [2,3] and W.Balser, B.L.J.Braaksma, J.-P.Ramis and Y.Sibuya [1]. In particular, B.L.J.Braaksma [3] treated nonlinear cases by means of a method based on J.Ecalle's theory of acceleration (cf. J.Ecalle [4] and J.Martinet and J.-P.Ramis [8]). In this paper, we shall outline another proof based on the cohomological definition of multisummability (cf. B.Malgrange and J.-P.Ramis [7] and W.Balser, B.L.J.Braaksma, J.-P.Ramis and Y.Sibuya [1]). The main problem is explained in §2 (cf. Theorem 2.1). In this paper, we shall outline a proof of Theorem 2.1 only, since multisummability of formal power series solutions can be derived from Theorem 2.1 in a manner similar to the proof of Theorem 4.1 based on Lemma 7.1 in paper [1]. In our outline, we shall show mostly the formal part which is the key idea. An analytic justification of the formal part utilizes methods due to M.Hukuhara [5], M.Iwano [6], and J.-P.Ramis and Y.Sibuya [10]. We shall publish another paper (jointly with J.-P.Ramis) in which the entire analysis will be explained in detail.