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Multipliers for certain convolution measure algebras


Author: Charles Dwight Lahr
Journal: Trans. Amer. Math. Soc. 185 (1973), 165-181
MSC: Primary 43A10
DOI: https://doi.org/10.1090/S0002-9947-1973-0333587-6
MathSciNet review: 0333587
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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0333587-6
Keywords: Convolution measure algebra, $ {l_1}$-algebra, multiplier
Article copyright: © Copyright 1973 American Mathematical Society

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