Geometric representation theory for unitary groups of operator algebras

Abstract

Geometric realizations for the restrictions of GNS representations to unitary groups of C * -algebras are constructed. These geometric realizations use an appropriate concept of reproducing kernels on vector bundles. To build such realizations in spaces of holomorphic sections, a class of complex coadjoint orbits of the corresponding real Banach-Lie groups is described and some homogeneous holomorphic Hermitian vector bundles that are naturally associated with the coadjoint orbits are constructed.

Geometric Perspectives on Reproducing Kernels

“…j / H K on spanfK W 2 Dg whose completion gives rise to a Hilbert space denoted by H K , which consists of sections of the bundle …; see [8,Th. 4.2].…”

“…Recently, reproducing kernels on vector fiber bundles (which are a clear geometric object) has revealed fruitful in representation theory and function theory [11,20,33] and to construct Borel-Weil-type (infinite-dimensional) realizations of GNS representations as well as Stinespring dilations [3,8]. In the present article, the attention is focused on the specific differential geometry aspects of such a relationship.…”

On Complex Infinite-Dimensional Grassmann Manifolds

“…Using a charaterization (and labelling) of invariant complex structures on infinite-dimensional homogeneous manifolds (Theorem 6.1 in [4]), we prove in Section 3 below that U A /U A (p) and G A /G A ([p]) are locally biholomorphic complex manifolds. Moreover, it is also shown that G A /G A (p) is a complexification of U A /U A (p), and then these results are translated in terms of homogeneous vector bundles, in the spirit of Theorem 5.8 in [5]. (A complementary perspective on these manifolds can be found in [6].)…”

Integrability of Analytic Almost Complex Structures on Banach Manifolds

“…This point is particularly important for the applications to the adjoint or coadjoint orbits that show up in connection with the operator algebras (see [9,12,13,19] and Corollary 16). Indeed, the unitary groups of C * -algebras are in general nonconnected (although they are connected in the special case of W * -algebras), while nothing seems to be known on connectedness of the unitary groups associated with general non-separable operator ideals (compare Remark 4.9 in [9]).…”