Jordan structures in harmonic functions and Fourier algebras on homogeneous spaces
type: Item
instance of: scholarly article
title: Jordan structures in harmonic functions and Fourier algebras on homogeneous spaces
zbMATH Open document ID: 1108.43009
author: Cho-Ho Chu
author: A. T.-M. Lau
published in: Mathematische Annalen
publication date: 2007-01-26T00:00:00Z
full work available at URL: http://cds.cern.ch/record/977052
review text: A Jordan triple system is a complex vector space \(V\) with a Jordan triple product \(\{.,.,.\}: V \times V \times V \to V\) which is symmetric and linear in the outer (the first and the third) variables, conjugate linear in the middle (second) variable and satisfies the Jordan triple identity \(\{a,b,\{x,y,z\}\} = \{\{a,b,c\},y,z\} -\{x,\{a,b,y\},z\} + \{z,y,\{a,b,z\}\}.\) On a \(C^*\)-algebra a triple product structure can be defined by \(\{x,y,z\} = \frac{1}{2}(xy^*z + zy^*c).\) Six types of JBW*-triples are well known as Cartan factors, e.g. the Cartan factor of type 1 is the Banach space \(\mathcal L(H,K)\) of bounded operators between complex Hilbert spaces \(H\) and \(K\) with Jordan triple product \(\{x,y,z\} = \frac{1}{2}(xy^*z + zy^*x).\) The authors prove that there is a contractive projection from the space \(L^\infty(\Omega,M_n)\) of essentially bounded matrix \(M_n\)-valued functions on \(\Omega\) with respect to the fixed quasi-invariant measure \(\nu\) on \(\Omega\) onto the space \(H_\sigma(\Omega,M_n)=\{ f\in L^\infty(\Omega,M_n) ; f= f\* \sigma \}\) for some \(M_n\)-valued measure \(\sigma\). From the text: ``Consequently, \(H_\sigma(\Omega,M_n)\) is a Jordan triple system and is, moreover, isometric to a finite \(\ell^\infty\)-sum \(\bigoplus_k L^\infty(\Omega_k \otimes C_k)\), where \(C_k\) is a finite-dimensional Cartan factor of type \(1,2,3\), or 4. It follows from the spectral result in Section 3 that the spectrum of \(H_\sigma(\Omega,M_n)\), regarded as Poisson space of \(H_\sigma(\Omega,M_n)\), is a finite disjoint union of Stonean spaces. We [i.e., the authors] give necessary and sufficient conditions for the coincidence of these two structures. We also show that \(H_\sigma(\Omega,M_n)\) is always a subalgebra of \(L^\infty(\Omega,M_n)\) whenever \(\Omega\) is a homogeneous space of a nilpotent group and \(\sigma\) is symmetric.''
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 46L70
Mathematics Subject Classification ID: 43A05
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 45E10
Mathematics Subject Classification ID: 31C05
zbMATH DE Number: 5118855
zbMATH Keywords: Jordan triple
zbMATH Keywords: Fourier algebras
reviewed by: Do Ngoc Diep
MaRDI profile type: Publication
OpenAlex ID: W2000082441
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DOI: 10.1007/S00208-006-0013-Y
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description: scientific article; zbMATH DE number 5118855