## Rocky Mountain Journal of Mathematics

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

Jaeseong Heo

#### 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

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

#### 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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