α-POSITIVE/α-NEGATIVE DEFINITE FUNCTIONS ON GROUPS
- Authors
- Heo, Jaeseong
- Issue Date
- 2018
- Publisher
- ROCKY MT MATH CONSORTIUM
- Keywords
- alpha-positive/alpha-negative definite function; Krein space; J-representation; J-cocycle; proper (alpha, J)-action
- Citation
- ROCKY MOUNTAIN JOURNAL OF MATHEMATICS, v.48, no.1, pp.249 - 268
- Indexed
- SCIE
SCOPUS
- Journal Title
- ROCKY MOUNTAIN JOURNAL OF MATHEMATICS
- Volume
- 48
- Number
- 1
- Start Page
- 249
- End Page
- 268
- URI
- https://scholarworks.bwise.kr/hanyang/handle/2021.sw.hanyang/150871
- DOI
- 10.1216/RMJ-2018-48-1-249
- ISSN
- 0035-7596
- 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 (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 (K, J) corresponding to a proper matricially bounded a-negative definite function.
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