[Paper Review] Representing multipliers of the Fourier algebra on non-commutative $L^p$ spaces

Matthew Daws|White Rose Research Online (University of Leeds, The University of Sheffield, University of York)|Jun 28, 2009

Advanced Operator Algebra Research30 references6 citations

TL;DR

This paper constructs a canonical isometric representation of the completely bounded multiplier algebra $ M_{cb}A(G) $ on a direct sum of non-commutative $ L^p $ spaces associated to the right von Neumann algebra of a locally compact group $ G $. It establishes that $ A_2(\hat{G}) $ is isometrically isomorphic to $ L^1(G) $, generalizing the abelian case, and introduces a new Figa-Talamanca–Herz-type algebra $ A_p(\hat{G}) $ built from these spaces, showing it contains a dense subalgebra of $ C^*_\lambda(G) $.

ABSTRACT

We show that the multiplier algebra of the Fourier algebra on a locally compact group $G$ can be isometrically represented on a direct sum on non-commutative $L^p$ spaces associated to the right von Neumann algebra of $G$. If these spaces are given their canonical Operator space structure, then we get a completely isometric representation of the completely bounded multiplier algebra. We make a careful study of the non-commutative $L^p$ spaces we construct, and show that they are completely isometric to those considered recently by Forrest, Lee and Samei; we improve a result about module homomorphisms. We suggest a definition of a Figa-Talamanca--Herz algebra built out of these non-commutative $L^p$ spaces, say $A_p(\hat G)$. It is shown that $A_2(\hat G)$ is isometric to $L^1(G)$, generalising the abelian situation.

Motivation & Objective

• To construct a canonical isometric representation of the multiplier algebra of the Fourier algebra $ A(G) $ on non-commutative $ L^p $ spaces associated to the right von Neumann algebra of $ G $.
• To extend the duality framework of Pontryagin and Young to non-abelian groups via non-commutative $ L^p $ spaces and complex interpolation.
• To define and study a new Figa-Talamanca–Herz-type algebra $ A_p(\hat{G}) $ built from these non-commutative $ L^p $ spaces.
• To show that $ A_2(\hat{G}) $ is canonically isometric to $ L^1(G) $, generalizing the abelian case.
• To prove that $ A_p(\hat{G}) $ contains a dense subset that forms a subalgebra of $ C^*_\lambda(G) $ for $ 1 < p < ∞ $.

Proposed method

• Use complex interpolation to construct non-commutative $ L^p $ spaces $ L^p(\hat{G}) $ associated to the right von Neumann algebra of $ G $, using weights and the dual group structure.
• Equip the direct sum $ E = \bigoplus_{n} L^{p_n}(\hat{G}) $ with the $ \ell^2 $-norm and canonical operator space structure for $ p_n \to 1 $.
• Define a representation $ \pi^p_* $ of $ A(G) \otimes_{\gamma} A(G) $ on $ \mathcal{B}(E) $ via the left regular representation and the operator $ K $, which intertwines $ L^2(G) $ with $ L^2(\hat{G}) $.
• Show that the image of $ \pi^p_* $ identifies with $ A_p(\hat{G}) $, and that $ \pi^p_* $ is isometric and weak*-weak* continuous.
• Use the operator space structure on $ E $ to define a completely bounded representation of $ M_{cb}A(G) $, showing it is completely isometric and weak*-weak* continuous.
• Prove that $ A_p(\hat{G}) $ contains a dense subalgebra of $ C^*_\lambda(G) $ by constructing elements via convolution and support arguments on $ C_{00}(G)^2 $.

Experimental results

Research questions

• RQ1 Can the multiplier algebra $ MA(G) $ be represented isometrically on a direct sum of non-commutative $ L^p $ spaces associated to $ VN(G) $?
• RQ2 Is there a natural generalization of the Figa-Talamanca–Herz algebra $ A_p(G) $ for non-abelian groups, defined via non-commutative $ L^p $ spaces?
• RQ3 Does $ A_2(\hat{G}) $ coincide with $ L^1(G) $ under the canonical isometry, extending the abelian case?
• RQ4 Is the algebra $ A_p(\hat{G}) $ itself a Banach algebra, or only a dense subalgebra of $ C^*_\lambda(G) $?
• RQ5 Can the module actions on $ L^p(\hat{G}) $ be realized as pointwise multiplication of functions, and are these spaces completely isometric to those in [10]?

Key findings

• The multiplier algebra $ MA(G) $ admits a weak*-weak* continuous isometric representation on the $ \ell^2 $-direct sum $ E = \bigoplus_n L^{p_n}(\hat{G}) $, with $ p_n \to 1 $.
• When $ E $ is equipped with the canonical operator space structure, $ M_{cb}A(G) $ embeds completely isometrically and weak*-weak* continuously into $ \mathcal{CB}(E) $.
• The space $ A_2(\hat{G}) $ is canonically isometric to $ L^1(G) $, with the quotient norm on $ A_2(\hat{G}) $ matching the $ L^1 $-norm on $ \lambda(L^1(G)) $.
• The algebra $ A_p(\hat{G}) $ contains a dense subset that forms a subalgebra of $ C^*_\lambda(G) $, constructed via convolution and support control on $ C_{00}(G)^2 $.
• The non-commutative $ L^p(\hat{G}) $ spaces constructed via complex interpolation are completely isometric to those defined by Forrest, Lee, and Samei, improving their results on module homomorphisms.
• The construction provides a natural framework to extend cohomological results from [10] to all $ p \in (1,\infty) $, not just $ p \geq 2 $.