Operator-valued measures, dilations, and the theory of frames (Q2925675)

In operator theory, Naimark's dilation theorem characterizes positive operator-valued measures (POVMs) as follows: Let \(E:\Sigma\to B(\mathcal{H})\) be a positive operator-valued measure. Then there exists a Hilbert space \(\mathcal{K}\), a bounded linear operator \(V:\mathcal{H}\to\mathcal{K}\), and an orthogonal projection-valued measure (OPVM) \(F:\Sigma\to B(\mathcal{K})\) such that \(E(B)=V^\ast F(B) V\). One of the primary objectives of this memoir is to provide extensions of Naimark's fundamental theorem to the case of non-Hilbertian normed Banach operator-valued measures. The main results are outlined in the introduction of this memoir as follows.NEWLINENEWLINETheorem A states that operator-valued measures (OVM) factor through projection-valued measures: For Banach spaces \(X,Y\), if \(E:\Sigma\to B(X,Y)\) is an operator-valued measure, then there is an intermediate Banach space \(Z\), bounded operators \(S:Z\to Y\) and \(T:X\to Z\) and a projection-valued probability measure \(F:\Sigma\to B(Z)\) such that \(E(B)=S\, F(B)\, T\). The system \((F,Z,S,T)\) is called a Banach dilation system (or Hilbert system if \(Z\) is a Hilbert space).NEWLINENEWLINETheorem B addresses properties of injective dilations. When the OVM \(E:\Sigma\to B(H)\) admits a Hilbert dilation \((E,\mathcal{H},S,T)\), Theorem C states that there exists a corresponding Hilbert dilation system \((F,\mathcal{K},V^\ast,V)\) such that \(V:\mathcal{H}\to\mathcal{K}\) is an isometric embedding.NEWLINENEWLINEA framing for a Banach space \(X\) is a pair of sequences \(\{x_i\}\subset X\) and \(\{y_i\}\subset X^\ast\) such that \(x=\sum \langle x, y_i\rangle x_i\) converges unconditionally for any \(x\in X\). Framings generalize frames to Banach spaces and they are scalable: if \(\{\alpha_i\}\subset\mathbb{C}\setminus\{0\}\), then \(\{\alpha_ix_i, y_i/\overline{\alpha}_i\}\) is a framing when \(\{x_i, y_i\}\) is. Theorem D states that if \(\{x_i,y_i\}\) is a framing for a Hilbert space \(\mathcal{H}\), then the OVPM \(E\) induced by this framing has a Hilbert space dilation if and only if there is a rescaling sequence \(\{\alpha_i\}\subset\mathbb{C}\) such that \(\{\alpha_i x_i\}\) and \(\{y_i/\overline{\alpha}_i\} \) are frames for \(\mathcal{H}\). In this case, \(E\) is a completely bounded map. This case occurs in particular when \(\inf \|x_i\|\|y_i\|>0\). Not all framings are rescalable to frames as Theorem E states: There exists a framing for a Hilbert space \(\mathcal{H}\) whose induced OVM \(E\) is not completely bounded, hence cannot be rescaled to a framing that admits a Hilbert space dilation.NEWLINENEWLINEThe remainder of the memoir focuses on systems of unitary operators. Corollary F (to Theorem D) states that if \(\mathcal{U}_1\) and \(\mathcal{U}_2\) are unitary systems on a separable Hilbert space \(\mathcal{H}\) (such as Gabor or wavelet systems) and if there exist \(x,y\in\mathcal{H}\) such that \(\{\mathcal{U}_1x,\mathcal{U}_2y\}\) is a framing for \(\mathcal{H}\), then \(\{\mathcal{U}_1x\}\) and \(\{\mathcal{U}_2y\}\) are both frames for \(\mathcal{H}\). Further results involve applications to Banach and von Neumann algebras. Theorem G states that if \(\mathcal{A}\) is a purely atomic abelian von Neumann algebra acting on a separable Hilbert space, then every ultra-weakly continuous linear map \(\phi:\mathcal{A}\to B(\mathcal{H})\) admits a Banach space \(Z\) and ultra-weakly continuous unital homomorphism \(\pi:\mathcal{A}\to B(Z)\), and bounded operators \(T:\mathcal{H}\to Z\) and \(S:Z\to\mathcal{H}\) such that \(\phi(a)=S\pi(a)T\) for all \(a\in\mathcal{A}\). The ideas in the proof of Theorem G provide the basis for a universal dilation theorem, Theorem H for bounded mapping between Banach algebras, which states that if \(\mathcal{A}\) is a Banach algebra, \(X\) is a Banach space, and \(\phi:\mathcal{A}\to B(X)\) is a bounded linear operator, then there exists a Banach space \(Z\), a bounded linear unital homomorphism \(\pi:\mathcal{A}\to B(Z)\), and bounded linear operators \(T:X\to Z\) and \(S:Z\to X\) such that \(\phi(a)=S\pi(a) T\) for all \(a\in\mathcal{A}\). Further new results for mappings of von Neumann algebras in the noncommutative case are also given which generalize particular cases of \textit{W. F. Stinespring}'s dilation theorem [Proc. Am. Math. Soc. 6, 211--216 (1955; Zbl 0064.36703)]. Standard discrete Hilbert space frame theory can be identified as the special case in which the domain algebra is abelian and purely atomic, the map is completely bounded, and the OVM is purely atomic and completely bounded with rank-1 atoms. Connections are also made with Kadison's similarity problem for bounded von Neumann algebra homomorphisms.