Abstract
In this chapter we are mainly interested in the study of abelian locally compact groups A, their dual groups \(\hat{A}\) together with various associated group algebras. Using the Gelfand-Naimark Theorem as a tool, we shall then give a proof of the Plancherel Theorem, which asserts that the Fourier transform extends to a unitary equivalence of the Hilbert spaces \(L^2(A)\) and \(L^2(\widehat A)\). We also prove the Pontryagin Duality Theorem that gives a canonical isomorphism between A and its bidual \({\hat{\hat{A}}}\).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Deitmar, A., Echterhoff, S. (2014). Duality for Abelian Groups. In: Principles of Harmonic Analysis. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-05792-7_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-05792-7_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-05791-0
Online ISBN: 978-3-319-05792-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)