Rocky Mountain Journal of Mathematics

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

Jaeseong Heo

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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

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

First available in Project Euclid: 28 April 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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