Advances in Operator Theory

$L^p$ Fourier transformation on non-unimodular locally compact groups

Marianne Terp

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Abstract

Let $G$ be a locally compact group with modular function $\Delta$ and left regular representation $\lambda$. We define the $L^p$ Fourier transform of a function $f \in L^p(G)$, $1 \le p \le 2$, to be essentially the operator $\lambda(f)\Delta^{\frac{1}{q}}$ on $L^2(G)$ (where $\frac{1}{p}+\frac{1}{q}=1$) and show that a generalized Hausdorff-Young theorem holds. To do this, we first treat in detail the spatial $L^p$ spaces $L^p(\psi_0)$, $1 \le p \le \infty$, associated with the von Neumann algebra $M=\lambda(G)^{\prime\prime}$ on $L^2(G)$ and the canonical  weight $\psi_0$ on its commutant. In particular, we discuss isometric isomorphisms of $L^2(\psi_0)$ onto $L^2(G)$ and of $L^1(\psi_0)$ onto the Fourier algebra $A(G)$. Also, we give a characterization of positive definite functions belonging to $A(G)$ among all continuous positive definite functions.

Article information

Source
Adv. Oper. Theory, Volume 2, Number 4 (2017), 547-583.

Dates
Received: 19 August 2017
Accepted: 19 September 2017
First available in Project Euclid: 4 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.aot/1512431728

Digital Object Identifier
doi:10.22034/AOT.1709-1231

Mathematical Reviews number (MathSciNet)
MR3730047

Zentralblatt MATH identifier
1374.39040

Subjects
Primary: 39B82: Stability, separation, extension, and related topics [See also 46A22]
Secondary: 44B20 46C05: Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)

Keywords
$L^p$ Fourier transformation locally compact group Fourier algebra positive definite function

Citation

Terp, Marianne. $L^p$ Fourier transformation on non-unimodular locally compact groups. Adv. Oper. Theory 2 (2017), no. 4, 547--583. doi:10.22034/AOT.1709-1231. https://projecteuclid.org/euclid.aot/1512431728


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