##
**Harmonic analysis of probability measures on hypergroups.**
*(English)*
Zbl 0828.43005

de Gruyter Studies in Mathematics. 20. Berlin: de Gruyter. vi, 601 p. (1995).

Hypergroups generalize locally compact groups. They appear when the Banach space of all bounded Radon measures on a locally compact space carries a convolution having all properties of a group convolution apart from the fact that the convolution of two point measures is a probability measure with compact support and not necessarily a point measure. Discrete hypergroups are closely related to Schur rings, Hecke algebras, and association schemes. While these objects are mainly studied under algebraic and number theoretic aspects, hypergroups were introduced by C. F. Dunkl [Trans. Am. Math. Soc. 179, 331–348 (1978; Zbl 0241.43003)], R. Jewett [Adv. Math. 18, 1–101 (1975; Zbl 0325.42017)], and R. Spector [Trans. Am. Math. Soc. 239, 147–166 (1978; Zbl 0428.43001)]. The intention was to unify harmonic analysis on duals of compact groups, double coset spaces \(G//H\) \((H\) a compact subgroup of a locally compact group \(G)\), and commutative convolution algebras associated with product linearization formulas of special functions. Recently it turned out that quantum Gelfand pairs as well as bifinite irreducible bimodules of \(\text{II}_1\)-factors also lead to hypergroups. Harmonic analysis and probability theory on commutative hypergroups are well developed meanwhile where many results from group theory remain valid. Besides monographs from Russian mathematicians on hypercomplex systems (which admit more complex axioms than hypergroups), the research monograph under review is the first one which deals with hypergroups from the point of view of harmonic analysis and probability theory.

The monograph of Bloom and Heyer is organized as follows: The introductory chapter discusses the foundations of hypergroups, their measure algebras as well as basic constructions which lead from other mathematical objects (like groups) to hypergroups. The highlight of chapter 1 is the existence and uniqueness of Haar measures due to Jewett and Spector.

Chapter 2 is devoted to dual spaces of commutative hypergroups. Many concepts presented here (like Fourier transform and Plancherel measure) carry over from the theory of locally compact abelian groups without substantial changes. On the other hand, some fundamental new problems arise here for commutative hypergroups: First of all, the dual space \(\widehat K\) of a commutative hypergroup \(K\) usually does not admit a canonical dual hypergroup structure (except for some classes of examples coming from group theory). Moreover, it may occur that the support \(\text{supp} \pi\) of the Plancherel measure \(\pi\) is a proper subset of \(\widehat K\); even worse, the trivial character may not be contained in \(\text{supp} \pi\). Defects of this kind sometimes lead to serious problems in algebraic probability theory. To overcome at least some of them, the authors describe the modification of hypergroup convolutions by means of positive characters in Section 2.3. This useful modification procedure has no counterpart in group theory.

Chapter 3 serves as a source of concrete examples of hypergroups; in particular, it illustrates that many special functions (like orthogonal polynomials, Jacobi functions, and so on) are closely related to concrete hypergroup structures. This shows that many seemingly only formally related formulas for many special functions admit natural common interpretations in the framework of hypergroups. Major parts of chapter 3 are devoted to polynomial hypergroups (i.e., discrete commutative hypergroups whose characters are orthogonal polynomials) and Sturm-Liouville hypergroups on compact intervals and on the half axis \([0, \infty[\). The latter are commutative hypergroups whose characters are eigenfunctions of a Sturm- Liouville differential operator. Chapter 3 closes with recent contributions of W. Connett and A. Schwartz as well as Hm. Zeuner to the classification of all one-dimensional hypergroups with additional nice properties.

Chapters 4–7 are concerned with probability theory on commutative hypergroups. Chapter 4 mainly contains some preparations from harmonic analysis. For instance, the authors there deal with positive and negative definite functions, the Lévy continuity theorem as well as a Lévy- Khintchine representation. These results are used in chapter 5 to investigate convolution semigroups, infinitely divisible measures, and factorization properties of the semigroup of all probability measures of a commutative hypergroup. The algebraic probability theory developed in chapters 4 and 5 is related to results discussed in other monographs on similar topics; see H. Heyer [Probability measures on locally compact groups. Berlin etc.: Springer Verlag (1977; Zbl 0376.60002)], K. R. Parthasarathy [Probability measures on metric spaces. New York etc.: Academic Press (1967; Zbl 0153.19101)], and I. Z. Ruzsa and G. J. Székely [Algebraic probability theory. Chichester (UK): Wiley (1988; Zbl 0653.60012)]. Chapter 6 contains contributions to transient convolution semigroups. Main topics are the Chung-Fuchs criterion, potential measures and invariant Dirichlet forms.

In chapter 7 the authors study random walks on hypergroups and their limit behaviour when time tends to infinity. In the group case, such random walks are defined as products \((X_1 \cdot X_2 \cdots X_n)_{n \geq 1}\) of independent group-valued random variables \(X_i\). This method can be transferred to hypergroups with the aid of a construction called concretization according to Hm. Zeuner. This fundamental construction is introduced in Section 7.1. The remaining sections contain limit theorems for random walks on polynomial hypergroups on \(\mathbb{N}_0\) and Sturm-Liouville hypergroups on \([0, \infty[\). Proofs of these strong laws of large numbers, central limit theorems, and invariance principles rely heavily on generalized moment functions and the asymptotic behaviour of the characters of the underlying moment functions and the asymptotic behaviour of the characters of the underlying hypergroup structures. An interesting aspect is that different growth properties of the underlying hypergroups lead to a completely different limit behaviour of the associated random walks.

The concluding chapter 8 provides an outlook to some recent developments of the theory of hypergroups.

The monograph closes with a fairly complete list of references with more than 600 items as well as a list of examples of hypergroups.

The monograph under review was written during a period of rapid development of parts of the theory; it attempts to cover most aspects of hypergroup theory: foundations, examples, harmonic analysis, algebraic probability theory, potential theory, and limit theorems for random walks. It is therefore clear that some topics cannot be treated in a completely satisfying and self-contained way in a single book. Sometimes, it is very useful for the reader to be familiar with the corresponding results from group theory, and in my opinion the paper of Jewett remains indispensible for a first introduction. On the other hand, this research monograph gives a useful comprehensive account to a rapidly growing field of research, and it therefore seems to be necessary that this useful monograph appears now. I expect that researchers from different mathematical disciplines will benefit from the principles presented in the book.

The monograph of Bloom and Heyer is organized as follows: The introductory chapter discusses the foundations of hypergroups, their measure algebras as well as basic constructions which lead from other mathematical objects (like groups) to hypergroups. The highlight of chapter 1 is the existence and uniqueness of Haar measures due to Jewett and Spector.

Chapter 2 is devoted to dual spaces of commutative hypergroups. Many concepts presented here (like Fourier transform and Plancherel measure) carry over from the theory of locally compact abelian groups without substantial changes. On the other hand, some fundamental new problems arise here for commutative hypergroups: First of all, the dual space \(\widehat K\) of a commutative hypergroup \(K\) usually does not admit a canonical dual hypergroup structure (except for some classes of examples coming from group theory). Moreover, it may occur that the support \(\text{supp} \pi\) of the Plancherel measure \(\pi\) is a proper subset of \(\widehat K\); even worse, the trivial character may not be contained in \(\text{supp} \pi\). Defects of this kind sometimes lead to serious problems in algebraic probability theory. To overcome at least some of them, the authors describe the modification of hypergroup convolutions by means of positive characters in Section 2.3. This useful modification procedure has no counterpart in group theory.

Chapter 3 serves as a source of concrete examples of hypergroups; in particular, it illustrates that many special functions (like orthogonal polynomials, Jacobi functions, and so on) are closely related to concrete hypergroup structures. This shows that many seemingly only formally related formulas for many special functions admit natural common interpretations in the framework of hypergroups. Major parts of chapter 3 are devoted to polynomial hypergroups (i.e., discrete commutative hypergroups whose characters are orthogonal polynomials) and Sturm-Liouville hypergroups on compact intervals and on the half axis \([0, \infty[\). The latter are commutative hypergroups whose characters are eigenfunctions of a Sturm- Liouville differential operator. Chapter 3 closes with recent contributions of W. Connett and A. Schwartz as well as Hm. Zeuner to the classification of all one-dimensional hypergroups with additional nice properties.

Chapters 4–7 are concerned with probability theory on commutative hypergroups. Chapter 4 mainly contains some preparations from harmonic analysis. For instance, the authors there deal with positive and negative definite functions, the Lévy continuity theorem as well as a Lévy- Khintchine representation. These results are used in chapter 5 to investigate convolution semigroups, infinitely divisible measures, and factorization properties of the semigroup of all probability measures of a commutative hypergroup. The algebraic probability theory developed in chapters 4 and 5 is related to results discussed in other monographs on similar topics; see H. Heyer [Probability measures on locally compact groups. Berlin etc.: Springer Verlag (1977; Zbl 0376.60002)], K. R. Parthasarathy [Probability measures on metric spaces. New York etc.: Academic Press (1967; Zbl 0153.19101)], and I. Z. Ruzsa and G. J. Székely [Algebraic probability theory. Chichester (UK): Wiley (1988; Zbl 0653.60012)]. Chapter 6 contains contributions to transient convolution semigroups. Main topics are the Chung-Fuchs criterion, potential measures and invariant Dirichlet forms.

In chapter 7 the authors study random walks on hypergroups and their limit behaviour when time tends to infinity. In the group case, such random walks are defined as products \((X_1 \cdot X_2 \cdots X_n)_{n \geq 1}\) of independent group-valued random variables \(X_i\). This method can be transferred to hypergroups with the aid of a construction called concretization according to Hm. Zeuner. This fundamental construction is introduced in Section 7.1. The remaining sections contain limit theorems for random walks on polynomial hypergroups on \(\mathbb{N}_0\) and Sturm-Liouville hypergroups on \([0, \infty[\). Proofs of these strong laws of large numbers, central limit theorems, and invariance principles rely heavily on generalized moment functions and the asymptotic behaviour of the characters of the underlying moment functions and the asymptotic behaviour of the characters of the underlying hypergroup structures. An interesting aspect is that different growth properties of the underlying hypergroups lead to a completely different limit behaviour of the associated random walks.

The concluding chapter 8 provides an outlook to some recent developments of the theory of hypergroups.

The monograph closes with a fairly complete list of references with more than 600 items as well as a list of examples of hypergroups.

The monograph under review was written during a period of rapid development of parts of the theory; it attempts to cover most aspects of hypergroup theory: foundations, examples, harmonic analysis, algebraic probability theory, potential theory, and limit theorems for random walks. It is therefore clear that some topics cannot be treated in a completely satisfying and self-contained way in a single book. Sometimes, it is very useful for the reader to be familiar with the corresponding results from group theory, and in my opinion the paper of Jewett remains indispensible for a first introduction. On the other hand, this research monograph gives a useful comprehensive account to a rapidly growing field of research, and it therefore seems to be necessary that this useful monograph appears now. I expect that researchers from different mathematical disciplines will benefit from the principles presented in the book.

Reviewer: Michael Voit (Tübingen)

### MSC:

43A62 | Harmonic analysis on hypergroups |

43-02 | Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis |

28A33 | Spaces of measures, convergence of measures |

20N20 | Hypergroups |

33C65 | Appell, Horn and Lauricella functions |

60B99 | Probability theory on algebraic and topological structures |

60F05 | Central limit and other weak theorems |

60F15 | Strong limit theorems |

33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |

31C99 | Generalizations of potential theory |