A review is given of recent work aimed at constructing a quantum theory of cosmology in which all observables refer to information measurable by observers inside the universe. At the classical level the algebra of observables should be modified to take into account the fact that observers can only give truth values to observables that have to do with their backwards light cone. The resulting al...

Symmetries of trigonometric integrable two dimensional statistical face models are considered. The corresponding symmetry operators on the Hilbert space of states of the quantum version of these models define a weak *-Hopf algebra isomorphic to the Ocneanu double triangle algebra(DTA). R. TRINCHERO 1

We develop a complete theory of non-formal deformation quantization on the cotangent bundle weakly exponential Lie group. An appropriate integral formula for star-product is introduced together with suitable space functions which well defined. This becomes Fréchet algebra as pre-C*-algebra. Basic properties are proved, and extension to Hilbert an distributions given. A C*-algebra observables st...

In this note we announce some new methods and results in the theory of nonnormal Hilbert space operators and nonself ad joint operator algebras. A main difficulty in the subject has been the apparent absence of relations between, say, a nonself ad joint algebra of operators and its generated C*-algebra. For example, given full information about the norm-closed algebra P(T) generated by all poly...

An affine Lie algebra acts on cohomology groups of quiver varieties of affine type. A Heisenberg algebra acts on cohomology groups of Hilbert schemes of points on a minimal resolution of a Kleinian singularity. We show that in the case of type A the former is obtained by Frenkel-Kac construction from the latter.

Let {Dn} be a sequence of bounded invertible operators on Hilbert space H. It is shown that the collection of operators T for which the norm-limit limDnTD n exists is an algebra. Furthermore, some sufficient conditions on this sequence are established for the corresponding algebra to have a nontrivial invariant subspace. By considering specific sequences of operators several invariant subspace ...

In this brief note (written as a lengthy letter), we describe the construction of a representation for the Weyl-algebra underlying Loop Quantum Geometry constructed from a diffeomorphism variant state, which corresponds to a ”condensate” of Loop Quantum Geometry, resembling a static spatial geometry. We present the kinematical GNS-representation and the gaugeand diffeomorphism invariant Hilbert...

Symmetries of trigonometric integrable two dimensional statistical face models are considered. The corresponding symmetry operators on the Hilbert space of states of the quantum version of these models define a weak *-Hopf algebra isomorphic to the Ocneanu double triangle algebra. R. TRINCHERO 1

The C*-algebra of bounded operators on the separable Hilbert space cannot be mapped to a W*-algebra in such a way that each unital commutative C*-subalgebra C(X) factors normally through `∞(X). Consequently, there is no faithful functor discretizing C*-algebras to W*-algebras this way.