α-positive/α-negative definite functions on groups

Research output: Contribution to journalArticle

Abstract

In this paper, we introduce the notions of an α-positive/α-negative definite function on a (discrete) group. We first construct the Naimark-GNS type representation associated to an α-positive definite function and prove the Schoenberg type theorem for a matricially bounded α-negative definite function. Using a J-representation on a Krein space (K, J) associated to a nonnegative normalized α-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 α-positive definite functions and proper (α, J)-actions on a Krein space (K, J) corresponding to a proper matricially bounded α-negative definite function.

Original languageEnglish
Pages (from-to)249-268
Number of pages20
JournalRocky Mountain Journal of Mathematics
Volume48
Issue number1
DOIs
StatePublished - 2018 Jan 1

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Krein Space
Positive Definite Functions
Cocycle
Representation Type
Discrete Group
Non-negative
Theorem

Keywords

  • J)-action.
  • J-cocycle
  • Jrepresentation
  • Krein space
  • Proper (α
  • Α-positive/α-negative definite function

Cite this

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abstract = "In this paper, we introduce the notions of an α-positive/α-negative definite function on a (discrete) group. We first construct the Naimark-GNS type representation associated to an α-positive definite function and prove the Schoenberg type theorem for a matricially bounded α-negative definite function. Using a J-representation on a Krein space (K, J) associated to a nonnegative normalized α-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 α-positive definite functions and proper (α, J)-actions on a Krein space (K, J) corresponding to a proper matricially bounded α-negative definite function.",
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α-positive/α-negative definite functions on groups. / Heo, Jaeseong.

In: Rocky Mountain Journal of Mathematics, Vol. 48, No. 1, 01.01.2018, p. 249-268.

Research output: Contribution to journalArticle

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