Vol. 21, No. 2, 1967

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The measure algebra of a locally compact semigroup

Arne P. Baartz

Vol. 21 (1967), No. 2, 199–214

Let G be a locally compact idempotent, commutative, topological semigroup (semi-lattice). Let (G) denote its measure algebra, i.e., (G) consists of all countably additive regular Borel-measures defined on G and has the usual Banach algebra structure: pointwise linear operations, convolution, and total variation norm. To understand the structure of such a convolution algebra one studies its maximal ideals, the nature of the Gelfand transform, the structure of the closed ideal and the related question of spectral synthesis, etc.

In this paper G is the carlesian product of topological semigroups Gα of the following form: Gα is a linearly ordered set, locally compact in ils order topology; multiplication in Gα is given by xy = max(x,y). The product semigroup is assumed locally compact in the product topology.

The main theorem of this paper gives a representation of the space of maximal ideals Δ(G), for a finite product, in terms of the dual semigroup Ĝ. The multiplicative linear functionals of (G) are integrals of fixed semi-characters

τ(μ ) = |eχ(x)dμ(x),μ ∈ M (G).

It is shown that this integral representation does not hold for infinite products because the semi-characters are usually not integrable.

Mathematical Subject Classification
Primary: 42.56
Received: 28 September 1965
Published: 1 May 1967
Arne P. Baartz