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2018 $\alpha $-positive/$\alpha $-negative definite functions on groups
Jaeseong Heo
Rocky Mountain J. Math. 48(1): 249-268 (2018). DOI: 10.1216/RMJ-2018-48-1-249

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.

Citation

Download Citation

Jaeseong Heo. "$\alpha $-positive/$\alpha $-negative definite functions on groups." Rocky Mountain J. Math. 48 (1) 249 - 268, 2018. https://doi.org/10.1216/RMJ-2018-48-1-249

Information

Published: 2018
First available in Project Euclid: 28 April 2018

zbMATH: 06866709
MathSciNet: MR3795742
Digital Object Identifier: 10.1216/RMJ-2018-48-1-249

Subjects:
Primary: 22D20, 43A35
Secondary: 16W22, 46K10

Rights: Copyright © 2018 Rocky Mountain Mathematics Consortium

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Vol.48 • No. 1 • 2018
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