\(C^*\)-algebras. Volume 2: Banach algebras and compact operators (Q2724097)

Vol. 2 of ``\(C^*\)-algebras'' by C.~Constantinescu builds, together with preparatory vol.1 (see the review above), the base of the project. In a logical order the general theory of Banach algebras is presented without invoking the rich \(C^*\)-structure studied in later volumes. The approach is typical for many books on \(C^*\)-algebras but here an accent on studying Banach algebras per se is stronger, cf. vol. 1 of ``Fundamentals of operator algebras'' by \textit{R.~Kadison} and \textit{J.~Ringrose} (1994; Zbl 0831.46060) giving a solid introduction. NEWLINENEWLINENEWLINEThe reviewed volume consists of ch. 2, Banach algebras, and ch. 3, Compact operators. These topics are developed in detail, with numerous useful examples, distinguishing the author's style; cf. the concise treatment of the subject in ch. 1 of ``\(C^*\)-algebras and operator theory'' by \textit{G. J.~Murphy} (1990; Zbl 0714.46041). Vol. 2 contains all kinds of results, except branch (8), according to subdivision into the main stem and based on it independent branches (1) Infinite matrices, (2) Banach categories, (3) Nuclear maps, (4) Locally compact groups, (5) Differential equations, (6) Laurent series, (7) Clifford algebras, (8) Hilbert \(C^*\)-modules, and supplements. The large ch. 2 has 4 sections: Algebras, Normed algebras, Involutive Banach algebras and Gelfand algebras. We only note that illuminating examples range from quaternions to convolution algebras associated with a l.c. group, complexification of (involutive) real algebras is studied thoroughly, interesting corrolaries of the resolvent analiticity preclude equality \([x,y]=1\) in a complex initial Banach algebra (Wielandt's theorem). Within branch (6) poles of resolvent are studied, relevant modules appear as a special case of Banach categories and illustrations to the Gelfand transform end with the Fourier transform in the Schwartz space. NEWLINENEWLINENEWLINECh. 3 providing applications of the theory of Banach algebras is considerably shorter. Sec. 3.1 deals with the general theory of compact operators between \(B\)-spaces, in particular, with their spectra. Fredholm operators and integral operators are studied. For example, Grothendieck's identification of nuclear operators between the conjugate \(L^p\) and \(L^q\)-spaces with integral operators arises. Sec. 3.2 devoted to linear differential equations has a subsection on solving boundary value problems (on a real segment) for ordinary differential operators of \(n\)th order via the Green function. Supplementary results involve the notion of adjoint operator. The last subsection considers elliptic operators on a Riemannian manifold \(T\) [see \textit{C. Constantinescu} and \textit{A. Cornea}, ``Potential theory of harmonic spaces'' (1972; Zbl 0248.31011)] In particular, the eigenvalues and eigenspaces of Laplacians over some special \(T\) are described. NEWLINENEWLINENEWLINEAs well as the first volume, this one will be useful for the mathematical community.