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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Semigroups with positive definite structure


Author: Parfeny P. Saworotnow
Journal: Proc. Amer. Math. Soc. 40 (1973), 421-425
MSC: Primary 43A35; Secondary 22A25
DOI: https://doi.org/10.1090/S0002-9939-1973-0320640-1
MathSciNet review: 0320640
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Abstract: Let $ G$ be a semigroup with the identity 1 and an involution $ x \to {x^ \ast }$. One can define a complex-valued and an $ {H^ \ast }$-algebra valued positive definite function on $ G$ in the obvious way. Assume that for each $ x \in G$ there exists a positive number $ {L_x}$ such that $ q({x^ \ast }x) \leqq {L_x}q(1)$ for each complex positive definite function on $ G$. It is shown that each $ {H^ \ast }$-algebra valued positive definite function on $ G$ is of the form $ p(x) = (f,{T_x}f)$ for some representation $ T$ of $ G$ on a Hilbert module $ H$ and $ f \in H$. Also there is an analogue of Bochner theorem for $ G$.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0320640-1
Article copyright: © Copyright 1973 American Mathematical Society