Baire Measures on Homogeneous Compact Hyper Groups

Neeta Mahesh Patil

Baire Measures on Homogeneous Compact Hyper Groups

by Neeta Mahesh Patil*,

- Published in Journal of Advances in Science and Technology, E-ISSN: 2230-9659

Volume 9, Issue No. 19, May 2015, Pages 0 - 0 (0)

Published by: Ignited Minds Journals

ABSTRACT

An infinite compact group is necessarilyuncountable, by the Baire category theorem. A compact "hyper group, inwhich the product of two points is a probability measure, is much like acompact group, having an everywhere supported invariant measure, an orthogonalsystem of characters which span the continuous functions in the uniformtopology, and a multiplicative semigroup of positive-definite functions. It isremarkable that a compact hyper group can be count ably infinite. In this paperhyper groups, which include the algebra of measures on the p-adic integerswhich are invariant under the action of the units (for p = 2, 3, 5, * * * ) ispresented and investigate the question of whether the spectrum or some subsetof it has a hyper group structure.

KEYWORD

Baire Measures, Homogeneous Compact Hyper Groups, compact group, Baire category theorem, product of two points, probability measure, invariant measure, orthogonal system of characters, continuous functions, uniform topology

INTRODUCTION

The basic theory of hyper groups has been developed. A hyper group is a compact space on which the space M (H) of (finite) regular Borel measures is a commutative Banach algebra under its natural norm, possessing a multiplication (denoted by *), and such that the space Mp (H) of probability measures is a compact commutative topological (jointly continuous multiplication) semigroup with unit under the weak- topology- a compact commutative topological semigroup with unit. A hyper group is a locally compact space on which the space of finite regular Borel measures has a commutative convolution structure preserving the probability measures (Dunkl, 1973. Dunkl, Ramirez, 1971. Received May 10, 1972. Hasse, Vorlesungen, 1950). The spectrum of the measure algebra of a locally compact abelian group is the semigroup of all continuous semi characters of a commutative compact topological semi group. In this paper we consider the spectrum of abstract measure algebra and investigate the question of whether the spectrum or some subset of it has a hyper group structure (René Spector, 1970. Dunkl, Ramirez, 1971).

REVIEW OF LITERATURE:

For a hyper group H there exists a continuous map defined by . For the space of continuous functions on H, and define , by if a hyper group H possesses an invariant measure , (that is, and a continuous involution such that, And, Then H is called a -hyper group (spt denotes the minimum closed subset of H carrying the measure). Consequences of these definitions are (l) the space H of characters is an orthogonal basis for L2 (H, dm), and (2) spt m = H. 1. Symmetrization of hyper groups: The method of Symmetrization of a hyper group was introduced by the authors in (Pym, 1968). We will in this paper use this construction to produce a denumerable compact F*-hyper group-a striking contrast to infinite compact groups. Given a homeomorphism t on a compact F*-hyper group //, define Tj: be the (weak-* continuous) ad joint of T1that is, the homeomorphism T is called an auto morphism if . 2. A countable compact hyper groups: Motivated by the results of §3, we will in this section show how to construct for any compact countable F*-hyper group. For p prime and a = 1/p the example agrees with the hyper group HW constructed in §3 (Glicksberg, 1959. Ragozin, 1972. Taylor, 1965). Let a be such that and define to be the compact space Define the measure m on by For each define,

CONCLUSION:

The theory is then applied to the measure algebra of a compact P*-hyper group, the algebra of central measures on a compact group, or the algebra of measures on certain homogeneous spaces. A further hypothesis, which is satisfied by the algebra of measures given by ultra-spherical series, is given and it is used to give a complete description of the spectrum and the idempotent in this case. In this paper we consider the spectrum of theoretical measure algebra and investigate the question of whether the spectrum or some subset of it has a hyper group structure

REFERENCES:

C. Dunkl and D. Ramirez, (1971). Topics in Harmonic Analysis, Apple ton-Century-Crofts, New York. C. F. Dunkl (1973). The measure algebra of a locally compact hypergroup. Trans. Amer. Math. Soc. 179, pp. 331-348. C. F. Dunkl and D. E. Ramirez (1971). Topics in harmonic analysis, Appleton-CenturyCrofts, New York.

212-229.

H. Hasse, Vorlesungen über Zahlentheorie (1950). Die Grundlehren der math. Wissenschaften, Band 59, Springer-Verlag, Berlin. MR 14, 534. I. Glicksberg (1959). Convolution semigroups of measures, Pacific J. Math., 9, pp. 51-67. J. L. Taylor (1965). The structure of convolution measure algebras, Trans. Amer. Math. Soc, 119, pp. 150-166. J. Pym, (1968). Weakly separately continuous measures algebras, Math. Annalen, 175. pp. 207-219. Received May 10, (1972). This research was partly supported by NSF Grant GP-31483X. René Spector, (1970) Sur la structure locale des groupes abéliens localement compacts, Bull. Soc. Math. France Suppl. Mém. 24, 94 pp. MR 44 #729.