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Proceedings of the American Mathematical Society

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



Generalized positive linear functionals on a Banach algebra with an involution

Author: Parfeny P. Saworotnow
Journal: Proc. Amer. Math. Soc. 31 (1972), 299-304
MathSciNet review: 0287321
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Abstract: Let $ A$ be a proper $ {H^\ast}$-algebra and let $ B$ be a Banach $ \ast$-algebra with an identity. A linear mapping $ \varphi :B \to A$ is called a positive $ A$-functional if $ {\Sigma _{i,j}}a_i^\ast\varphi (x_i^\ast{x_j}){a_j}$ is positive for all $ {x_1},{x_2}, \cdots ,{x_n} \in B$ and $ {a_1},{a_2}, \cdots ,{a_n} \in A$. It is shown that for each positive $ A$-functional $ \varphi $ there exists a $ \ast$-representation $ T$ of $ B$ by $ A$-linear operators on a Hilbert module $ H$ such that $ \varphi (x) = ({f_0},Tx{f_0})$ for all $ x \in B$ and some $ {f_0} \in H$. If $ B$ is of the form $ B = \{ \lambda e + x\vert\lambda $ complex, $ e$ is the (abstract) identity, $ x \in {L^1}(G)\} $ for some locally compact group $ G$ then $ \varphi $ has the form $ \varphi (\lambda e + x) = \lambda \varphi (e) + \int {{\text{ }}_G}x(t)p(t)dt$ for some generalized ($ A$-valued) positive definite function $ p$ on $ G,x \in {L^1}(G)$.

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Keywords: Hilbert module, $ {H^\ast}$-algebra, Banach $ \ast$-algebra, group algebra, $ \ast$-representation
Article copyright: © Copyright 1972 American Mathematical Society