Rocky Mountain Journal of Mathematics

$\alpha $-positive/$\alpha $-negative definite functions on groups

Jaeseong Heo

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Abstract

In this paper, we introduce the notions of an $\alpha $-positive/$\alpha $-negative definite function on a (discrete) group. We first construct the Naimark-GNS type representation associated to an $\alpha $-positive definite function and prove the Schoenberg type theorem for a matricially bounded $\alpha $-negative definite function. Using a $J$-representation on a Krein space $(\mathcal{K} ,J)$ associated to a nonnegative normalized $\alpha $-negative definite function, we also construct a $J$-cocycle associated to a $J$-representation. Using a $J$-cocycle, we show that there exist two sequences of $\alpha $-positive definite functions and proper $(\alpha ,J)$-actions on a Krein space $(\mathcal{K} ,J)$ corresponding to a proper matricially bounded $\alpha $-negative definite function.

Article information

Source
Rocky Mountain J. Math., Volume 48, Number 1 (2018), 249-268.

Dates
First available in Project Euclid: 28 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1524880890

Digital Object Identifier
doi:10.1216/RMJ-2018-48-1-249

Mathematical Reviews number (MathSciNet)
MR3795742

Zentralblatt MATH identifier
06866709

Subjects
Primary: 22D20: Representations of group algebras 43A35: Positive definite functions on groups, semigroups, etc.
Secondary: 16W22: Actions of groups and semigroups; invariant theory 46K10: Representations of topological algebras with involution

Keywords
$\alpha $-positive/$\alpha $-negative definite function Krein space $J$-representation $J$-cocycle proper $(\alpha ,J)$-action

Citation

Heo, Jaeseong. $\alpha $-positive/$\alpha $-negative definite functions on groups. Rocky Mountain J. Math. 48 (2018), no. 1, 249--268. doi:10.1216/RMJ-2018-48-1-249. https://projecteuclid.org/euclid.rmjm/1524880890


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