The formalism of local coordinates on infinite-dimensional Hilbert manifolds is introduced. Images of charts on the manifolds are allowed to belong to arbitrary Hilbert spaces of functions including spaces of generalized functions. The corresponding local coordinate form of algebra of tensor fields on Hilbert manifolds is constructed. A single tensor equation in the formalism is shown to produc...

The paper contains constructions of Hilbert systems for the action of the circle group T using subgroups of implementable Bogoljubov unitaries w.r.t. Fock representations of the Fermion algebra for suitable data of the selfdual framework: H is the reference Hilbert space, Γ the conjugation and P a basis projection on H. According to a general result for Hilbert systems of this type, the group C...

We introduce the notion of automorphic symbol generalizing the classical modular symbol and use it to attach very general p-adic L-functions to nearly ordinary Hilbert automorphic forms. Then we establish an exact control theorem for the p-adically completed cohomology of a Hilbert modular variety localized at a suitable nearly ordinary maximal ideal of the Hecke algebra. We also show its freen...

Frames on Hilbert C*-modules have been defined for unital C*algebras by Frank and Larson [5] and operator-valued frames on a Hilbert space have been studied in [8]. The goal of this paper is to introduce operatorvalued frames on a Hilbert C*-module for a σ-unital C*-algebra. Theorem 1.4 reformulates the definition given in [5] in terms of a series of rank-one operators converging in the strict ...

Usually in quantum mechanics the Heisenberg algebra is generated by operators of position and momentum. The algebra is then represented on an Hilbert space of square integrable functions. Alternatively one generates the Heisenberg algebra by the raising and lowering operators. It is then natural to represent it on the Bargmann Fock space of holomorphic functions. In the following I show that th...

We investigate the structure of the Schrödinger algebra and its representations in a Fock space realized in terms of canonical Appell systems. Generalized coherent states are used in the construction of a Hilbert space of functions on which certain commuting elements act as self-adjoint operators. This yields a probabilistic interpretation of these operators as random variables. An interesting ...

We study GIT semistability of Hilbert points of Milnor algebras of homogeneous forms. Our first result is that a homogeneous form F in n variables is GIT semistable with respect to the natural SL(n)-action if and only if the gradient point of F, which is the first non-trivial Hilbert point of the Milnor algebra of F, is semistable. We also prove that the induced morphism on the GIT quotients is...

We study Kähler functions on the Hilbert ball and and the algebra of Kähler functions. We introduce quantum mechanics and operator algebras on the Hilbert ball by using the theory of geometrical quantum mechanics.

Characterization of the *-subalgebras in the algebra of bounded operators acting on Hilbert space is presented. Sufficient conditions for the existence of a faithful representation in pre-Hilbert space of a *-algebra in terms of its Groebner basis are given. These conditions are generalization of the unshrinkability of monomial *-algebras introduced by C. Lance and P. Tapper. The applications t...

We compute the C*-algebra generated by a group of composition operators acting on certain reproducing kernel Hilbert spaces over the disk, where the symbols belong to a non-elementary Fuchsian group. We show that such a C*-algebra contains the compact operperators, and its quotient is isomorphic to the crossed product C*-algebra determined by the action of the group on the boundary circle. In a...