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Nehari and Carathéodory-Fejér type extension results for operator-valued functions on groups

Author: Mihály Bakonyi
Journal: Proc. Amer. Math. Soc. 131 (2003), 3517-3525
MSC (2000): Primary 43A17, 47A57, 43A35, 47A20
Published electronically: February 20, 2003
MathSciNet review: 1991764
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Abstract: Let $G$ be a compact abelian group having the property that its character group $\Gamma $ is totally ordered by a semigroup $P$. We prove that every operator-valued function $k$ on $G$ of the form $k(x)=\sum\limits_{\gamma \in (-P)}\gamma (x)k_{\gamma }$, such that the Hankel operator $H_k$ is bounded, has an essentially bounded extension $K$ with $\vert\vert K\vert\vert _{\infty }=\vert\vert H_k\vert\vert$. The proof is based on Arveson's Extension Theorem for completely positive functions on $C^*$-algebras. Among the corollaries we have a Carathéodory-Fejér type result for analytic operator-valued functions defined on such groups.

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Additional Information

Mihály Bakonyi
Affiliation: Department of Mathematics, Georgia State University, Atlanta, Georgia 30303-3083

Received by editor(s): March 6, 2002
Received by editor(s) in revised form: June 16, 2002
Published electronically: February 20, 2003
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2003 American Mathematical Society