Hecke C*-algebras and semi-direct products

Steven Kaliszewski; Magnus B. Landstad; John Quigg

doi:10.1017/S0013091506001469

Hecke C*-algebras and semi-direct products

Steven Kaliszewski, Magnus B. Landstad, John Quigg

Mathematical and Statistical Sciences, School of (SoMSS)

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

We analyse Hecke pairs (G,H) and the associated Hecke algebra H when G is a semi-direct product N ⋊ Q and H = M R for subgroups M N and R Q with M normal in N. Our main result shows that, when (G,H) coincides with its Schlichting completion and R is normal in Q, the closure of Hin C*(G) is MoritaRieffel equivalent to a crossed product IQ/R, where I is a certain ideal in the fixed-point algebra C*(N)^R. Several concrete examples are given illustrating and applying our techniques, including some involving subgroups of GL(2,K) acting on K², where K = or K = [p¹]. In particular we look at the ax + b group of a quadratic extension of K.

Original language	English (US)
Pages (from-to)	127-153
Number of pages	27
Journal	Proceedings of the Edinburgh Mathematical Society
Volume	52
Issue number	1
DOIs	https://doi.org/10.1017/S0013091506001469
State	Published - Feb 2009

Keywords

Group C*-algebra
Hecke algebra
Morita equivalence
Semi-direct product

ASJC Scopus subject areas

General Mathematics

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10.1017/S0013091506001469

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title = "Hecke C*-algebras and semi-direct products",

abstract = "We analyse Hecke pairs (G,H) and the associated Hecke algebra H when G is a semi-direct product N ⋊ Q and H = M R for subgroups M N and R Q with M normal in N. Our main result shows that, when (G,H) coincides with its Schlichting completion and R is normal in Q, the closure of Hin C*(G) is MoritaRieffel equivalent to a crossed product IQ/R, where I is a certain ideal in the fixed-point algebra C*(N)R. Several concrete examples are given illustrating and applying our techniques, including some involving subgroups of GL(2,K) acting on K2, where K = or K = [p1]. In particular we look at the ax + b group of a quadratic extension of K.",

keywords = "Group C*-algebra, Hecke algebra, Morita equivalence, Semi-direct product",

author = "Steven Kaliszewski and Landstad, {Magnus B.} and John Quigg",

note = "Funding Information: Acknowledgements. We thank Arizona State University, the Norwegian University of Science and Technology and the Norwegian Science Foundation, who have all supported this research. We are also grateful to the anonymous referee for suggesting many improvements.",

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T1 - Hecke C*-algebras and semi-direct products

AU - Kaliszewski, Steven

AU - Landstad, Magnus B.

AU - Quigg, John

N1 - Funding Information: Acknowledgements. We thank Arizona State University, the Norwegian University of Science and Technology and the Norwegian Science Foundation, who have all supported this research. We are also grateful to the anonymous referee for suggesting many improvements.

PY - 2009/2

Y1 - 2009/2

N2 - We analyse Hecke pairs (G,H) and the associated Hecke algebra H when G is a semi-direct product N ⋊ Q and H = M R for subgroups M N and R Q with M normal in N. Our main result shows that, when (G,H) coincides with its Schlichting completion and R is normal in Q, the closure of Hin C*(G) is MoritaRieffel equivalent to a crossed product IQ/R, where I is a certain ideal in the fixed-point algebra C*(N)R. Several concrete examples are given illustrating and applying our techniques, including some involving subgroups of GL(2,K) acting on K2, where K = or K = [p1]. In particular we look at the ax + b group of a quadratic extension of K.

AB - We analyse Hecke pairs (G,H) and the associated Hecke algebra H when G is a semi-direct product N ⋊ Q and H = M R for subgroups M N and R Q with M normal in N. Our main result shows that, when (G,H) coincides with its Schlichting completion and R is normal in Q, the closure of Hin C*(G) is MoritaRieffel equivalent to a crossed product IQ/R, where I is a certain ideal in the fixed-point algebra C*(N)R. Several concrete examples are given illustrating and applying our techniques, including some involving subgroups of GL(2,K) acting on K2, where K = or K = [p1]. In particular we look at the ax + b group of a quadratic extension of K.

KW - Group C-algebra

KW - Hecke algebra

KW - Morita equivalence

KW - Semi-direct product

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DO - 10.1017/S0013091506001469

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JO - Proceedings of the Edinburgh Mathematical Society

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IS - 1

ER -

Hecke C*-algebras and semi-direct products

Abstract

Keywords

ASJC Scopus subject areas

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