[Paper Review] Tannaka-Krein duality for compact groupoids II, Fourier transform

Massoud Amini|ArXiv.org|Aug 27, 2003

Advanced Operator Algebra Research3 references3 citations

TL;DR

This paper extends Tannaka-Krein duality to compact groupoids by developing a non-abelian Fourier transform and proving a Plancherel theorem for $L^2$-functions on the groupoid. It establishes an isometric isomorphism between $L^2( ilde{rak{G}})$ and $L^2(rak{G})$, where $ ilde{rak{G}}$ is the dual space of irreducible representations, and proves a diagonal version of the Plancherel theorem for central functions on the isotropy groupoid.

ABSTRACT

In a series of papers, we have shown that from the representation theory of a compact groupoid one can reconstruct the groupoid using the procedure similar to the Tannaka-Krein duality for compact groups. In this part we study the Fourier and Fourier-Plancherel transforms and prove the Plancherel theorem for compact groupoids. We also study the central functions in the algebra of square integrable functions on the isotropy groups.

Motivation & Objective

• To generalize the Fourier and Plancherel theorems from compact groups to compact groupoids.
• To define a Fourier transform on the Banach algebra bundle $L^1(rak{G})$ and its extension to $L^2(rak{G})$.
• To establish a surjective isometric isomorphism between $L^2(rak{G})$ and $L^2(\hat{\frak{G}})$, the dual space of irreducible representations.
• To study central functions and central elements in $L^2(\frak{G})$, particularly on the isotropy groupoid $\frak{G}'$.
• To prove a diagonal version of the Plancherel theorem for the space of conjugacy classes ${\frak{G}'}^{\frak{G}}$.

Proposed method

• Define the Fourier transform $\mathfrak{F}_{u,v}(f)(\pi) = \int_{\mathcal{G}_u^v} f(x) \pi(x^{-1}) d\lambda_u^v(x)$ for $f \in L^1(\mathcal{G}_u^v)$, mapping to $\mathcal{B}(\mathcal{H}_v^\pi, \mathcal{H}_u^\pi)$.
• Construct the global Fourier transform $\mathfrak{F}(f)$ as a continuous section in $C_0(\hat{\frak{G}}, \mathcal{B}(\mathcal{H}))$ vanishing at infinity.
• Define the inverse Fourier transform $\check{g}$ for $g \in C_c(\hat{\frak{G}})$ via $\check{g}(x) = \sum_{\pi \in \hat{\frak{G}}} d_u^\pi g(\pi) \overline{\chi_v^\pi(x)}$.
• Equip $C_c(\hat{\frak{G}})$ with an inner product $\langle g,h \rangle = \sum_{\pi} (d_u^\pi)^2 \overline{g(\pi)} h(\pi)$ to define a Hilbert space structure.
• Prove that the inverse Fourier transform $\check{}$ extends to an isometry from $L^2(\hat{\frak{G}})$ to $L^2({\frak{G}'}^{\frak{G}})$, the space of square-integrable functions on the conjugacy groupoid.
• Characterize central elements in $L^2(\frak{G})$ as functions invariant under conjugation, and show $L^2({\frak{G}'}^{\frak{G}}) \simeq \mathfrak{C}L^2(\frak{G}')$.

Experimental results

Research questions

• RQ1 How can the Fourier transform be generalized from compact groups to compact groupoids?
• RQ2 What is the structure of the dual space $\hat{\frak{G}}$ of irreducible representations, and how does it relate to the conjugacy groupoid $\frak{G}^{\frak{G}}$?
• RQ3 Can the Plancherel theorem be extended to compact groupoids, and what is its form in the case of central functions?
• RQ4 What is the role of the spectral measure $d(\{\pi\}) = (d_u^\pi)^2$ in the duality and isometry?
• RQ5 How do the inverse Fourier transform and the Plancherel isometry behave on the subalgebra of central functions?

Key findings

• The Fourier transform $\mathfrak{F}_{u,v}(f)$ is defined as an integral operator mapping $f \in L^1(\mathcal{G}_u^v)$ to $\mathcal{B}(\mathcal{H}_v^\pi, \mathcal{H}_u^\pi)$, forming a continuous section in $C_0(\hat{\frak{G}}, \mathcal{B}(\mathcal{H}))$.
• The inverse Fourier transform $\check{g}$ for $g \in C_c(\hat{\frak{G}})$ satisfies $\check{g} * f = f * \check{g}$, showing it commutes with convolution.
• The inverse Fourier transform $\check{}$ extends uniquely to a bijective linear isometry from $L^2(\hat{\frak{G}})$ to $L^2({\frak{G}'}^{\frak{G}})$, the space of square-integrable functions on the conjugacy groupoid.
• The diagonal Plancherel theorem holds: $\hat{}: L^2({\frak{G}'}^{\frak{G}}) \to L^2(\hat{\frak{G}})$ is the inverse of $\check{}$, and $\langle \check{g}, \check{h} \rangle = \langle g, h \rangle$.
• The space of central elements $\mathfrak{C}L^2(\frak{G})$ is isometrically isomorphic to $L^2({\frak{G}'}^{\frak{G}})$, with the isomorphism induced by the canonical projection $q: \frak{G}' \to {\frak{G}'}^{\frak{G}}$.
• The spectral measure $d(\{\pi\}) = (d_u^\pi)^2$ is used to define the inner product on $C_c(\hat{\frak{G}})$, ensuring the Plancherel isometry preserves norms.