Introduction

Spectral Theories aim at studying the properties of certain normed algebras, the fundamental examples being the algebra of continuous functions with complex values on a topological space, the endomorphism algebra of a Banach space and the convolution algebra of integrable functions with respect to a Haar measure on a locally compact group. The scope of this study is not necessarily apparent from this simple indication, but it is only necessary, in order to grasp its importance, to observe that two contemporary mathematicians, unless they belong to the same school of thought, will very often have, in addition to linear algebra and the elementary part of differential and integral calculus, only certain aspects of Fourier analysis as common knowledge.

In the first two chapters of this Book, we develop the most fundamental ideas of spectral theories, and consider in greater detail the third example mentioned above, namely, the Fourier transform, in the setting of a commutative locally compact group.

The following chapters will first continue the general theory of compact linear mappings, then that of operators on Hilbert spaces, this case including linear mappings defined only on a subspace (in general dense) of such a space. As is known, this theory is the basis of the mathematical formalism of Quantum Mechanics. Finally, we shall present the elementary theory of unitary representations of topological groups, in particular compact groups, which we have already had occasion to use in chapter IX of the Book Lie Groups and Lie Algebras.

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The titles of the first two chapters are the same as those of the first edition of this Book, published in 1967. They have been revised, corrected and enlarged.

For ease of reading, we indicate below the most important changes which occur in this edition, and in particular the additions.

For chapter I:

– In n$^o3$ of §3, we introduce the notion of a proper partial mapping, which makes it possible to state in functorial form the Gelfand correspondence between locally compact spaces and commutative stellar algebras.

– In §4, we have added n$^{os}13$ and 14, which deal with the holomorphic functional calculus in real or complex normable algebras, and in algebras without a unit element.

– In §6, the presentation of the continuous functional calculus has been revised (n$^o6$). n$^{os}9$ to 11 are new; they consider the notion of a positive element in a general stellar algebra, the approximate units of such algebras, and the quotient of a stellar algebra by a closed two-sided ideal.

– §7, concerning the most elementary spectral properties of endomorphisms of Banach spaces, is essentially new; the numbers n$^{os}7$ and 8 partially take up certain numbers of the former paragraph 6. It is this new paragraph which will serve as the basis for the study of endomorphisms of Hilbert spaces in chapter IV.

For chapter II:

– §1 concerning the Fourier transform has been revised in detail, for example n$^o6$ which deals with the functoriality of Pontryagin duality. In the new n$^o9$, we make more explicit the most important statements for the groups $\mathbf{R}^n$ and $(\mathbf{R}/\mathbf{Z})^n$. Fourier theory in $\mathbf{R}^n$ finding its most natural setting in the theory of tempered distributions, we state its essential properties without giving the proofs, which will appear in chapter IV.

– We have finally added a Fourier theory form which summarizes the definitions and properties of the principal functional spaces involved in this theory, as well as the essential formulae.

Many exercises are new. One will note in particular the considerable increase in the number of exercises in §1 of chapter II, which reflects the mathematical importance of Fourier theory.

In the preceding books, the following reference to TS, to appear, must be modified as follows:

LIE, IX, p. 72, §7, n$^o4$. Instead of “d’après TS”, read “d’après TS, cor. du th. 1

de II, p. 215.”