Multipliers for certain convolution measure algebras

Lahr, Charles Dwight

doi:10.1090/S0002-9947-1973-0333587-6

Multipliers for certain convolution measure algebras
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by Charles Dwight Lahr PDF

Trans. Amer. Math. Soc. 185 (1973), 165-181 Request permission

Abstract:

Let ($A,\ast$) be a commutative semisimple convolution measure algebra with structure semigroup $\Gamma$, and let S denote a commutative locally compact topological semigroup. Under the assumption that A possesses a weak bounded approximate identity, it is shown that there is a topological embedding of the multiplier algebra $\mathcal {M}(A)$ of A in $M(\Gamma )$. This representation leads to a proof of the commutative case of Wendel’s theorem for $A = {L_1}(G)$, where G is a commutative locally compact topological group. It is also proved that if ${l_1}(S)$ has a weak bounded approximate identity of norm one, then $\mathcal {M}({l_1}(S))$ is isometrically isomorphic to ${l_1}(\Omega (S))$, where $\Omega (S)$ is the multiplier semigroup of S. Likewise, if S is cancellative, then $\mathcal {M}({l_1}(S))$ is isometrically isomorphic to ${l_1}(\Omega (S))$. An example is provided of a semigroup S for which ${l_1}(\Omega (S))$ is isomorphic to a proper subset of $\mathcal {M}({l_1}(S))$.

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