Hilbert-substructure of Real Measurable Spaces on Reductive Groups, I; Basic Theory (original) (raw)

On unitary representations of disconnected real reductive groups

arXiv (Cornell University), 2018

Let G be the real reductive group and let G 0 be the identity component. Let us assume that the unitary dual G 0 is known. In this paper (in Section 5) the unitary dual G is constructed. Automorphisms of G 0 generated by elements of G are the main ingredient of the construction. If the automorphism is outer, one has to consider the corresponding intertwining operators S. Operators S and their properties are analyzed in Section 4. Automorphisms of g 0 are closely related to automorphisms of G 0. They are investigated in Section 3. Automorphisms of so(4,4) are analyzed in Subsection 3.1.

The full Bochner theorem on real reductive groups

The major results of [3.], leading to the spherical Bochner theorem and its (spherical) extension, were made possible through the spherical transform theory of Trombi-Varadarajan [14.] and were greatly controlled by the non-availability of the full (non-spherical) Harish-Chandra Fourier transform theory on a general connected semisimple Lie group, G. Sequel to the recently announced results of Oyadare [13.], where the full image of the Schwartz-type algebras,

Invariant subspaces and harmonic conjugation on compact abelian groups

Pacific Journal of Mathematics, 1992

Let Γ be a dense subgroup of the real line 1. Endow Γ with the discrete topology and the order it inherits from R, and let K be the dual group of Γ. Helson's classic theory of generalized analyticity uses the spectral decomposability of unitary groups to establish a oneto-one correspondence between the cocycles on K and the normalized simply invariant subspaces of L 2 (K). This theory has been extended to the invariant subspaces of LP(K), 1 < p < oo, by using recent results concerning the spectral decomposability of uniformly bounded one-parameter groups acting on UMD spaces. We show here that each cocycle A on K can be used to transfer the classical Hubert transform from L ι (R) to L ι (K) in terms of almost everywhere convergence on K so that in the interesting case (i.e., when A is not a coboundary) the corresponding invariant subspace of LP{K) is a generalized ergodic Hardy space. This description of the invariant subspaces explicitly identifies the role of the Hubert transform in generalized analyticity on K. The formulation in terms of almost everywhere convergence on K provides an intrinsic viewpoint which extends to the case p = 1.

Spaces invariant under unitary representations of discrete groups

Journal of Mathematical Analysis and Applications, 2020

We investigate the structure of subspaces of a Hilbert space that are invariant under unitary representations of a discrete group. We work with square integrable representations, and we show that they are those for which we can construct an isometry intertwining the representation with the right regular representation, that we call a Helson map. We then characterize invariant subspaces using a Helson map, and provide general characterizations of Riesz and frame sequences of orbits. These results extend to the nonabelian setting several known results for abelian groups. They also extend to countable families of generators previous results obtained for principal subspaces.

Representations of groups with ordered duals and generalized analyticity

Journal of Functional Analysis, 1990

Let 6' be a locally compact abelian group whose dual group 6 has a Haar-measurable order. We show that for each strongly contmuous, umformly bounded representation R of G in a UMD space A'. thcrc is a corresponding dircctsum decomposition of X which reflects the order in 6. The projections in A' corresponding to this dtrect-sum decomposition have norms controlled solely by the bound of R and a constant depending only on X. We illustrate how this "vectorvalued harmonic conjugation" result generalizes the various abstract successors of the M. Riesz theorem and we introduce an application to the superdiagonalizatton of kernels for abstract integral operators.

Unitary representations of real reductive groups

2012

The basic idea of our algorithm is to follow the behavior of the signature of the Hermitian form through this deformation, counting changes through singularities of the form at reducibility points. An important technical tool is replacing the classical invariant form (in which the real form of the Lie algebra acts by skew-adjoint operators) by forms in which the compact form of the Lie algebra acts by skew-adjoint operators.

Banach spaces related to integrable group representations and their atomic decompositions, I

Journal of Functional Analysis, 1989

We present a general theory of Banach spaces which are invariant under the action of an integrable group representation and give their atomic decompositions with respect to coherent states, i.e., the atoms arise from a single element under the group action. Several well-known decomposition theories are contained as special examples and are unified under the aspect of group theory.

Inner Product Groups and Riesz Representation Theorem

Symmetry, 2021

In this paper, we introduce an inner product on abelian groups and, after investigating the basic properties of the inner product, we first show that each inner product group is a torsion-free abelian normed group. We give examples of such groups and describe the norms induced by such inner products. Among other results, Hilbert groups, midconvex and orthogonal subgroups are presented, and a Riesz representation theorem on divisible Hilbert groups is proved.