Geometric representation theory for unitary groups of operator algebras
The article under review provides geometric realizations for unitary representations of the unitary group \(U_A\) of a \(C^*\)-algebra~\(A\), which are restrictions of certain Gelfand-Naimark-Segal (GNS-) representations of~\(A\). Given a state \(\phi\colon A\to {\mathbb C}\), let \(\rho\colon A\to {\mathcal B}({\mathcal H}_\phi)\) be the GNS-representation associated with~\(\phi\). If \(\phi\circ E=\phi\) for some unital sub-\(C^*\)-algebra \(B\subseteq A\) and conditional expectation \(E\colon A\to B\), the authors show that \(\rho| _{U_A}\) can be realized as a representation on a reproducing kernel Hilbert space of continuous sections of a homogeneous Hermitian vector bundle \(D_{\phi,B}\) over \(U_A/U_B\). Stronger conclusions are obtained for \(A\) a \(W^*\)-algebra, if \(B=\{b\in A\colon (\forall a\in A)\phi(ab)=\phi(ba)\}\) and \(\phi(a)=\tau(ca)\) for some faithful normal tracial state \(\tau\colon A\to{\mathbb C}\) and \(c\in A\) such that \(0\leq c\), \(\tau(c)=1\) and the spectrum of~\(c\) is a finite set. Then \(U_A/U_B\) can be considered as a complex manifold (identifying it with a suitable homogeneous space of the unit group~\(A^\times\)), and \(D_{\phi,B}\) as a holomorphic vector bundle. Moreover, \(\rho| _{U_A}\) can be realized as a representation on a Hilbert space of holomorphic sections in \(D_{\phi,B}\), with a holomorphic reproducing kernel. As preliminaries, the appropriate versions of homogeneous vector bundles are discussed as well as reproducing kernel Hilbert spaces of continuous (or holomorphic) sections in Hermitian (possibly infinite-dimensional) vector bundles and the corresponding reproducing kernels. An exposition of reproducing kernel spaces in finite-dimensional vector bundles had been given earlier by [\textit{W. Bertram} and \textit{J. Hilgert}, pp.\ 43--58 in: \textit{H.-D. Doebner} et al.\ (editors), Lie theory and its applications in physics II, World Scientific, Singapore (1998)].
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