Structure semigroups of convolution measure algebras



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A brief outline is given for obtaining the structure semigroup S(M) of a convolution measure algebra M, in which the underlying topological space is the maximal ideal space of the norm closure of the linear span of AM, the complex homomorphisms on M. The multiplication in the structure semigroup of M is shown to be induced, in a natural manner, by the Arens product on M**. We obtain two covariant functors: The functor S from the category of convolution measure algebras and C-homomorphisms into the category of compact Abelian semigroups, and continuous homomorphisms, and the functor [omega] from the category of locally compact Abelian semigroups and continuous homomorphisms into the category of compact Abelian semigroups and continuous homomorphisms. If W is a compact semigroup. whose continuous semicharacters separate the points of W, then W is shown to be the continuous homomorphic image of omega. If N is an L-subalgebra of M, we obtain necessary and sufficient conditions to obtain the structure semigroup of N as a closed subsemigroup of S(M). We prove that each prime L-subalgebra induces a decomposition of S(M) into the disjoint union of a closed subsemigroup, the structure semigroup of N, and a closed ideal, the structure semigroup of N[squared]. Certain subgroups of a locally compact Abelian semigroup W are shown to have their Bohr compactifications as closed subsemigroups of omega. We prove the result: If each of W[lowered 1] and W[lowered 2] is a locally compact Abelian semigroup with identity then [Omega](W[lowered 1])x[omega](W[lowered 2])is a structure semigroup.