A family of countably compact 𝑃_{∗}-hypergroups

Abstract

An infinite compact group is necessarily uncountable, by the Baire category theorem. A compact <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Subscript asterisk"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>P</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{P_\ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-hypergroup, in which the product of two points is a probability measure, is much like a compact group, having an everywhere supported invariant measure, an orthogonal system of characters which span the continuous functions in the uniform topology, and a multiplicative semigroup of positive-definite functions. It is remarkable that a compact <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Subscript asterisk"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>P</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{P_\ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-hypergroup can be countably infinite. In this paper a family of such hypergroups, which include the algebra of measures on the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-adic integers which are invariant under the action of the units (for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p equals 2 comma 3 comma 5 comma midline-horizontal-ellipsis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>5</mml:mn> <mml:mo>,</mml:mo> <mml:mo>⋯<!-- ⋯ --></mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p = 2,3,5, \cdots )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) is presented. This is an example of the symmetrization technique. It is possible to give a nice characterization of the Fourier algebra in terms of a bounded-variation condition, which shows that the usual Banach algebra questions about the Fourier algebra, such as spectral synthesis, and Helson sets have easily determinable answers. Helson sets are finite, each closed set is a set of synthesis, the maximal ideal space is exactly the underlying hypergroup, and the functions that operate are exactly the Lip 1 functions.