Square-integrable imprimitivity systems (Q2738086)

Let \(X\) denote a locally compact second countable (lcsc) topological space and \(G\) a lcsc topological group. Assume that \(G\) has an action on \(X\) denoted by \(g[x]\). For a function \(f\) defined on \(X\) \(l_gf(x)=f(g^{-1}[x])\) defines an operator on \(C_0(X)\), the continuous functions on \(X\) which vanish at infinity. Definition: A couple \((M,U)\) is called an imprimitivity system (for \(G\) based on \(X)\) acting in a Hilbert space \(H\) if:NEWLINENEWLINENEWLINE(1) \(M\) is a nontrivial representation of \(C_0(X)\) in \(H\).NEWLINENEWLINENEWLINE(2) \(U\) is a representation of \(G\) in \(H\).NEWLINENEWLINENEWLINE(3) \(U_gM(f)U^{-1}_g=M(l_gf)\) for all \(g\) in \(G\), \(f\) in \(C_0(X)\).NEWLINENEWLINENEWLINELet \(\mu\) be a canonical measure on \(G\times X\) associated with the measures \(dg\) and \(dx\). \((M,U)\) is square-integrable if there exist nonzero \(u,v\) in \(H\) and \(c_{u,v}\) in \(L^2(G\times X,\mu)\) so that \(\langle u,U(\varphi)v\rangle=\int c_{u,v}\varphi(g,x)d\mu(g,x)\) for all \(\varphi\) compactly supported on \(G\times X\). -- The authors characterize general square integrable imprimitivity systems in terms of irreducible invariant closed subspaces of \(L^2(G\times X,\mu)\). They also give characterizations in the cases when \(G\) acts transitively on \(X\) and when \(X\) is an abelian lcsc group. Finally, the authors give examples which relate the Gabor analysis and windowed Fourier transforms on abelian groups to square-integrable imprimitivity systems. Of particular interest is the special unitary group \(G=SU(2)\) and \(X=3\)-dimensional sphere.