Jordan structures in harmonic functions and Fourier algebras on homogeneous spaces
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.''
scientific article; zbMATH DE number 5118855
scientific article