Geometric representation theory for unitary groups of operator algebras
type: Item
instance of: scholarly article
title: Geometric representation theory for unitary groups of operator algebras
zbMATH Open document ID: 1108.22008
author: Daniel Beltiţă
author: T. S. Ratiu
published in: Advances in Mathematics
publication date: 2006-12-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0501057
review text: 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)].
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 22E65
Mathematics Subject Classification ID: 43A65
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 46E22
Mathematics Subject Classification ID: 46K10
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L10
Mathematics Subject Classification ID: 46L30
Mathematics Subject Classification ID: 58B12
zbMATH DE Number: 5078973
zbMATH Keywords: Unitary group
zbMATH Keywords: unitary representation
zbMATH Keywords: reproducing kernel
zbMATH Keywords: space of sections
zbMATH Keywords: homogeneous vector bundle
zbMATH Keywords: geometric realization
zbMATH Keywords: operator algebra
zbMATH Keywords: Banach algebra
zbMATH Keywords: von Neumann algebra
zbMATH Keywords: GNS construction
MaRDI profile type: Publication
OpenAlex ID: W2085279325
arXiv ID: math/0501057
cites work: Manifolds, tensor analysis, and applications.
cites work: Q2723192
cites work: Q4798345
cites work: Extensions of pure states and projections of norm one
cites work: Extensions of Pure States and Projections of Norm One II
cites work: Theory of Reproducing Kernels
cites work: Subalgebras of \(C^*\)-algebras. III: Multivariable operator theory
cites work: Spectra for solvable Lie algebras of bundle endomorphisms
cites work: Integrability of analytic almost complex structures on Banach manifolds
cites work: Q5704404
cites work: Symplectic leaves in real Banach Lie-Poisson spaces
cites work: Q5557748
cites work: Q5655596
cites work: Representation theory of the Hilbert-Lie group U(H)//2
cites work: Representation Theory of U 1 (H)
cites work: Q3139085
cites work: Factor representations of diffeomorphism groups
cites work: The pure state space of quantum mechanics as Hermitian symmetric space
cites work: Representations of Hermitian kernels by means of Krein spaces. II. Invariant kernels
cites work: On L.\ Schwartz's boundedness condition for kernels
cites work: Q4857207
cites work: A Bott-Borel-Weil theory for direct limits of algebraic groups
cites work: Q4369283
cites work: Q4780257
cites work: Unitary operators in \(C^ *\)-algebras
cites work: Q5601136
cites work: Q4371512
cites work: Q2778938
cites work: The Bott-Borel-Weil theorem for direct limit groups
cites work: Representation theory and convexity
cites work: Q4793258
cites work: Q4670626
cites work: Banach Lie-Poisson spaces and reduction
cites work: Factorization problems for nests: Factorization methods and characterizations of the universal factorization property
cites work: Q4040547
cites work: Q5624220
cites work: Sous-espaces d'espaces vectoriels topologiques et noyaux associés. (Noyaux reproduisants.)
cites work: Theory of operator algebras. II
cites work: On the projection of norm one in $W^ *$-algebras
cites work: Q5186182
cites work: Principal series representations of direct limit groups
DOI: 10.1016/J.AIM.2006.02.009
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