Banach algebras and the general theory of \(^*\)-algebras. II: \(^*\)-algebras (Q2721514)

This is the second volume of the monumental work ''that provides a modern account of the basic Banach algebras theory including all known results on general Banach \(^*\)-algebras''. In spite of the size of the whole project (1617 pages in both volumes) the author excluded here two last chapters which were planned initially: Chapter 13 on the cohomology of Banach algebras and Chapter 14 on K-theory of functional Banach algebras and general Banach algebras. [For Volume I (1994) see Zbl 0809.46052].NEWLINENEWLINENEWLINEThe volume under review consists of a preface, four chapters (9-12) and the bibliograhy, which extends the bibliography of the first volume and counts 2179 positions. In size the second volume is larger than the first one, and counts 868 pages. The chapters 9-11 contain some author's results which were not formally published, but many of them were presented during numerous lectures. Chapter 12 is an expanded version of a survey article published by the author. NEWLINENEWLINENEWLINEChapter 9, entitled ``\(^*\)-Algebras'', consists of 9 sections. After preliminary remarks and examples in 9.1, the following section 9.2 is devoted to \(^*\)-representations of \(^*\)-algebras (we call them representations and algebras in the sequel). Here the representation space is either a Hilbert or a pre-Hilbert (incomplete) space. A representation \(T\) of an algebra \(A\) on a Hilbert space \(H\) is called essential if the set \(T_AH=\{T_ax:a\in A\), \(x\in H\}\) is dense in \(H\). In this case \(T_ax=0\) for all \(a\in A\) implies \(x=0\). If \(T\) is an arbitrary representation of an algebra \(A\) on a Hilbert space \(H\), the space \(H\) is an orthogonal direct sum of \(T\)-invariant subspaces \(H_T=(T_AH)^-\) (the closure) and \(H_0=\bigcap_{a\in A} \operatorname {ker} T_a\), and the restriction of \(T\) to \(H_T\) is essential. For this reason all later considered representations are assumed to be essential. Every essential representation is the direct sum of topologically cyclic representations. Further results in this sections concern representations of \(C^*\)-algebras, group algebras, and of the Calkin algebra. NEWLINENEWLINENEWLINESection 9.3 contains some information about von Neumann algebras and \(W^*\)algebras (von Neumann and Kaplansky density theorems, factors, the Borel functional calculus for normal elements) and the author himself calls this section ``a digression. Section 9.4 is devoted to topologically cyclic \(^*\)-representations. It starts with the relations between representable positive functionals on A and its representations on some \(H\) (the Gelfand-Naimark theory). It gives further a detailed description of various types of positive functionals and states, together with their relation to representations. NEWLINENEWLINENEWLINEIn Section 9.5 ``\(C^*\)-seminorms'' it is shown that every such seminorm on \(A\) is given by \(a\rightarrow \|T_a\|\) for a certain representation \(T\). Any \(C^*\)-algebra \(A\) has only one \(C^*\)-norm and this norm can be determined by the \(^*\)-structure of \(A\). Theorem 9.5.9 and Corollary 9.5.10 (due to the author) describe the situation when a Banach space \((A,\|\cdot\|)\) can be made into a unital \(C^*\)-algebra with the norm \(\|\cdot\|\). Further results concern the automatic submultiplicativity for \(C^*\)-norms (the Sebestien's result stating that the \(C^*\) condition implies submultiplicativity), geometric properties of \(C^*\)-norms, and states on \(C^*\)-algebras. More results on topologically irreducible representations are given in section 9.6. NEWLINENEWLINENEWLINESection 9.7 is entitled ``The Reducing Ideal and \(^*\)-Representation topology''. The ideal in question, denoted by \(A_R\), is the intersection of the kernels of all representations of \(A\) (as before, representation means a \(^*\)-representation) and \(A\) is said reduced if \(A_R=(0)\). The \(^*\)-representation topology on \(A\) is the weakest topology that makes each representation \(T\rightarrow B(H)\) continuous with respect to the norm topology on \(B(H)\). This is an m-convex topology and it is Hausdorff iff \(A\) is reduced. Some results concerning properties of finite-dimensional reduced algebras are gathered in a table given in section 9.1. Section 9.8 is entitled ``Hermitian and Symmetric \(^*\)-algebras''. Again some results concerning the finite-dimensional situation are given in the above mentioned table. NEWLINENEWLINENEWLINEThe last section of this chapter is devoted to linear maps between \(^*\)-algebras. The author considers here hermitian, positive, completely positive, operator completely positive, and Schwarz maps \(\varphi\) (the latter means \(\varphi (a)^*\varphi (a)\leq \varphi (a^*a)\)). He considers also some projection maps called expectations or quasi-expectations (under some assumptions a projection is a positive quasi-expectation iff it is Schwarz). One of major results here (due to Stinespring) states that a map from a commutative \(C^*\)-algebra to a \(C^*\)-algebra is completely positive iff it is positive. The section is concluded with Jordan \(^*\)-homomorphisms. NEWLINENEWLINENEWLINEAs the author states the ``Chapter 10 is the heart of this volume. Along with parts of Chapter 2 it is the central contributor of the whole two volume work. Section 10.5 on \(^*\)-regular \(BG^*\)-algebras is its newest portion.'' The chapter is entitled ``Special \(^*\)-algebras'' and consists of 6 sections. ``In this chapter we study various essentially algebraic hypotheses on \(^*\)-algebras, most of which are satisfied by Banach \(^*\)-algebras and all of which are satisfied by hermitian Banach \(^*\)-algebras. Virtually all known results on Banach \(^*\)-algebras and hermitian Banach \(^*\)-algebras (that are not explicitly properties of the complete norm) are obtained in this more general setting. We study these classes of \(^*\)-algebras partly for their own interest, but mainly because they lend themselves to particularly simple proofs of the theorems we wish to establish. Furthermore, we can define categories which include all Banach \(^*\)-algebras among their objects but which are much better behaved than the awkward categories of Banach \(^*\)-algebras.'' NEWLINENEWLINENEWLINEThe categories in questions consist of \(G^*\)-algebras and \(BG^*\)-algebras (introduced by the author in 1971 and 1973, but this is the first systematic account of their theory), they are defined in section 10.1, which gives basic properties of these algebras and their representations. Let \(A\) be (as before) a \(^*\)-algebra, it is said to be a \(G^*\)-algebra if \(\gamma (a)=\sup\{\|T_a\|: T\) is a \(^*\)-representation of \(A\) on a Hilbert space\} is finite for all \(a\) in \(A\). An algebra \(A\) (recall our convention: all algebras are \(^*\)-algebras) is said to be a \(BG^*\)-algebra if every of its representations on a pre-Hilbert space is normed. Every \(BG*\)-algebra is a \(G^*\)-algebra. An algebra \(A\) belongs to the class \(G^*\) iff it has the largest \(C^*\)-seminorm, called here the Gelfand-Naimark seminorm, and this seminorm equals to the quantity \(\gamma\) defined above. NEWLINENEWLINENEWLINEIn sections 10.2 and 10.3 the author introduces further classes of \(^*\)-algebras, contained in the class \(G^*\) and called \(T^*\)-algebras, \(S^*\)-algebras, \(Sq^*\)-algebras and \(U^*\)-algebras. The following algebras are in all these classes: Banach \(^*\)-algebras, ideals in Banach \(^*\)-algebras, functional \(^*\)-algebras, pseudo-Banach \(^*\)-algebras. In section 10.4 the author presents a generalization of the theory of hermitian Banach \(^*\)-algebras. To this end he introduces \(\gamma S^*\)-algebras defined as \(G^*\)-algebras in which the Gelfand-Naimark seminorm is spectral. This class is contained in the earlier introduced class \(S^*\) and the section gives the first full publication of the author's results. The remaining sections of this chapter are devoted to the ideal theory of \(^*\)-algebras. NEWLINENEWLINENEWLINEChapter 11, entitled ``Banach \(^*\)-Algebras'', consists of 7 sections. The first one summarizes the most basic results on Banach \(^*\)-algebras and their representations from previous chapters and adds further details. Other sections are devoted to the unitary structure of Banach \(^*\)-algebras, automatic continuity for Banach \(^*\)-algebras, hermitian Banach \(^*\)-algebras, ideal structure of Banach \(^*\)-algebras, Banach \(^*\)-algebras with minimal ideals and \(H^*\)-algebras, Hilbert algebras and Tomita algebras. NEWLINENEWLINENEWLINEThe last chapter 12 is entitled ``Locally Compact Groups and their \(^*\)-algebras. The first three sections give an introduction to locally compact groups (including the Iwasawa decomposition) and the fourth section gives an introduction to group representations. In the latter section the author is primarily interested in three Banach algebras associated with a locally compact group \(G\): \(L^1(G)\), \(M(G)\) and \(C^*(G)\). In the last two sections the author defines and studies 22 important classess of locally compact groups each of which contains all abelian groups and all compact groups. Examples: \([Z]\) -- the class of all central groups (\(G\) modulo its center is compact), \([SIN]\) -- the class of all groups with arbitrarily small neighbourhoods of the identity invariant under inner automorphisms, [Moore] -- the class of groups \(G\) for which all continuous, unitary, topological irreducible representations are finite-dimesional. Other considered classes consist of all unimodular groups, polynomial growth groups, distal groups, amenable groups, Tauberian groups or Wiener groups. Many properties of these classes are given in form of a table.