In a previous paper we defined the concept of an affinized projective variety and its associated Hilbert series. We computed the Hilbert series for varieties associated to quadratic monomial ideals. In this paper we show how to apply these results to affinized flag varieties. We discuss various examples and conjecture a correspondence between the Hilbert series of an affinized flag variety and ...

We describe the classical Schwinger model as a study of the pro-jective modules over the algebra of complex-valued functions on the sphere. On these modules, classified by π 2 (S 2), we construct her-mitian connections with values in the universal differential envelope which leads us to the Schwinger model on the sphere. The Connes-Lott program is then applied using the Hilbert space of complex...

In this paper we find a simple rule to reproduce the algebra of quantum observables using only the commutators and operators which appear in the Koopman-von Neumann (KvN) formulation of classical mechanics. The usual Hilbert space of quantum mechanics becomes embedded in the KvN Hilbert space: in particular it turns out to be the subspace on which the quantum positions Q and momenta P act irred...

With every E0–semigroup (acting on the algebra of of bounded operators on a separable infinite-dimensional Hilbert space) there is an associated Arveson system. One of the most important results about Arveson systems is that every Arveson system is the one associated with an E0–semigroup. In these notes we give a new proof of this result that is considerably simpler than the existing ones and a...

We show that every (continuous) faithful product system admits a (continuous) faithful nondegenerate representation. For Hilbert spaces this is equivalent to Arveson’s result that every Arveson system comes from an E0–semigroup. We point out that for Hilbert modules this is not so. As applications we show a C–algebra version of a result for von Neumann algebras due to Arveson and Kishimoto, and...

We present a general way to define a topology on orthomodular lattices. We show that in the case of a Hilbert lattice, this topology is equivalent to that induced by the metrics of the corresponding Hilbert space. Moreover, we show that in the case of a boolean algebra, the obtained topology is the discrete one. Thus, our construction provides a general tool for studying orthomodular lattices b...

We give an elementary proof of a result which characterizes onto *-isomorphisms of the algebra BL(H) of all the bounded linear operators on a Hilbert space H. A known proof of this result (Arveson, 1976) relies on the theory of irreducible representations of C∗-algebras, whereas the proof given by us is based on elementary properties of operators on a Hilbert space which can be found in any int...

The properties of deductive systems in Hilbert algebras are treated. If a Hilbert algebra H considered as an ordered set is an upper semilattice then prime deductive systems coincide with meet-irreducible elements of the lattice DedH of all deductive systems on H and every maximal deductive system is prime. Complements and relative complements of DedH are characterized as the so called annihila...

We determine the ideal structure of the Toeplitz C∗-algebra on the bidisk 0 Introduction A large part of doing research in mathematics is asking the right question. Posing a timely question can trigger thought and provoke insights, even when the matter has no good resolution. That is what happened in the case at hand. After devoting much time to the study of Hilbert modules and quotient Hilbert...

We show that the image of a commutative monotone sequentially complete C∗-algebra, under a sequentially normal morphism, is again a monotone sequentially complete C-algebra, and also a monotone sequentially closed C∗-subalgebra. As a consequence, the image of an algebra of this type, under a sequentially normal representation in a separable Hilbert space, is strongly closed. In the case of a un...