##
**Lipschitz algebras.**
*(English)*
Zbl 0936.46002

Singapore: World Scientific. xiii, 223 p. (1999).

This monograph is a systematic account of the theory of classical Lipschitz function spaces. For a metric space \(X\) and a fixed element \(e\) of \(X\), the Lipschitz space \(\text{Lip}_0(X)\) is the set of all scalar-valued Lipschitz functions on \(X\) which vanish at \(e\) with the standard algebraic operations and with the norm equal Lipschitz number, \(\text{Lip}_0(X)\) is a Banach space, and it has natural algebra and lattice structures as well. The diameter of the metric space \(X\) is assumed to be finite in most cases. The space \(\text{Lip}(X)\) of all bounded scalar-valued Lipschitz functions is also considered; but the author shows that, in an abstract sense, the spaces \(\text{Lip}(X)\) are special cases of the spaces \(\text{Lip}_0(X)\). Then little Lipschitz spaces \(\text{lip}_0(X)\) and \(\text{lip}(X)\) of functions from \(\text{Lip}_0(X)\) and \(\text{Lip}(X)\) satisfying the conditions
\[
|f(p)- f(q)|= o(\rho(p, q))\quad (\rho(p, q)\to 0)
\]
are considered.

The book contains 7 chapters. Chapter 1 is introductory. In Chapter 2 the spaces \(\text{Lip}_0(X)\) are considered as Banach spaces. The space \(\text{Lip}_0(X)\) is a dual Banach space, and the author gives the explicit construction of the predual in the form of the Arens-Eells space \(\text{AE}(X)\) [R. F. Arens and J. Eells jun., Pac. J. Math. 6, 397-403 (1956; Zbl 0073.39601)] (\(\text{AE}(X)\) is a completion of the metric space of finite-supported signed measures with total value \(0\) with Kantorovic-Rubinstein metric). Extreme points of the unit ball of \(\text{Lip}_0(X)^*\) are characterized. Versions of the Banach-Stone theorem for \(\text{Lip}(X)\) and \(\text{Lip}_0(X)\) are proved. Chapter 3 is devoted to the special case when \(\text{Lip}_0(X)\) is a double dual space (the metric space \(X\) is assumed to be compact). If \(\text{Lip}_0(X)\) separates points uniformly then \(\text{lip}_0(X)\) is a double predual of \(\text{Lip}_0(X)\). Chapter 4 is devoted to the algebraic properties of Lipschitz spaces. Here order complete subalgebras and ideals of \(\text{Lip}(X)\) and \(\text{lip}(X)\) are characterized; under some natural assumptions subalgebras and quotients by order complete ideals are themselves Lip or lip spaces; point derivations and the spectral synthesis problem are considered. Chapter 5 deals with the lattice structure of the unit ball of \(\text{Lip}(X)\). This unit ball is a completely distributive lattice. The converse is true: every normed vector lattice whose unit ball is a completely distributive lattice is isomorphic to some \(\text{Lip}(X)\). In Chapter 6 measurable metric spaces are defined. The definition uses distances between subsets rather than points. Lipschitz functions on the measurable metric space \((X,\mu)\) are defined. Subalgebras and lattice properties of \(\text{Lip}(X,\mu)\) are considered. Chapter 7 is devoted to derivations of functions in Lipschitz spaces (“On the unit interval, the Lipschitz condition is literally equivalent to bounded differentiability”). The author obtains some analogue of this assertion for the general Lipschitz algebras. Boundedness of derivations plays the role of bounded differentiability. Connections with noncommutative geometry of A. Connes [“Géométrie noncommutative” (1990; Zbl 0745.46067), Engl. transl. (1994; Zbl 0818.46076)] are sketched.

The main body of the text is well supplemented with numerous open problems.

In general the book is of interest to all specialists in the field of functional analysis; as well it can be served as a special course for post-graduate students.

The book contains 7 chapters. Chapter 1 is introductory. In Chapter 2 the spaces \(\text{Lip}_0(X)\) are considered as Banach spaces. The space \(\text{Lip}_0(X)\) is a dual Banach space, and the author gives the explicit construction of the predual in the form of the Arens-Eells space \(\text{AE}(X)\) [R. F. Arens and J. Eells jun., Pac. J. Math. 6, 397-403 (1956; Zbl 0073.39601)] (\(\text{AE}(X)\) is a completion of the metric space of finite-supported signed measures with total value \(0\) with Kantorovic-Rubinstein metric). Extreme points of the unit ball of \(\text{Lip}_0(X)^*\) are characterized. Versions of the Banach-Stone theorem for \(\text{Lip}(X)\) and \(\text{Lip}_0(X)\) are proved. Chapter 3 is devoted to the special case when \(\text{Lip}_0(X)\) is a double dual space (the metric space \(X\) is assumed to be compact). If \(\text{Lip}_0(X)\) separates points uniformly then \(\text{lip}_0(X)\) is a double predual of \(\text{Lip}_0(X)\). Chapter 4 is devoted to the algebraic properties of Lipschitz spaces. Here order complete subalgebras and ideals of \(\text{Lip}(X)\) and \(\text{lip}(X)\) are characterized; under some natural assumptions subalgebras and quotients by order complete ideals are themselves Lip or lip spaces; point derivations and the spectral synthesis problem are considered. Chapter 5 deals with the lattice structure of the unit ball of \(\text{Lip}(X)\). This unit ball is a completely distributive lattice. The converse is true: every normed vector lattice whose unit ball is a completely distributive lattice is isomorphic to some \(\text{Lip}(X)\). In Chapter 6 measurable metric spaces are defined. The definition uses distances between subsets rather than points. Lipschitz functions on the measurable metric space \((X,\mu)\) are defined. Subalgebras and lattice properties of \(\text{Lip}(X,\mu)\) are considered. Chapter 7 is devoted to derivations of functions in Lipschitz spaces (“On the unit interval, the Lipschitz condition is literally equivalent to bounded differentiability”). The author obtains some analogue of this assertion for the general Lipschitz algebras. Boundedness of derivations plays the role of bounded differentiability. Connections with noncommutative geometry of A. Connes [“Géométrie noncommutative” (1990; Zbl 0745.46067), Engl. transl. (1994; Zbl 0818.46076)] are sketched.

The main body of the text is well supplemented with numerous open problems.

In general the book is of interest to all specialists in the field of functional analysis; as well it can be served as a special course for post-graduate students.

Reviewer: P.Zabreiko (Minsk)

### MSC:

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

46E25 | Rings and algebras of continuous, differentiable or analytic functions |

46E15 | Banach spaces of continuous, differentiable or analytic functions |