The common structure of the space of pure states P of a classical or a quantum mechanical system is that of a Poisson space with a transition probability. This is a topological space equipped with a Poisson structure, as well as with a function p : P × P → [0, 1], with certain properties. The Poisson structure is connected with the transition probabilities through unitarity (in a specific formu...

Let A be a Banach algebra and M be a Banach left A-module. A linear map δ : M → M is called a generalized derivation if there exists a derivation d : A → A such that δ(ax) = aδ(x) + d(a)x (a ∈ A,x ∈ M). In this paper, we associate a triangular Banach algebra T to Banach A-module M and investigate the relation between generalized derivations on M and derivations on T . In particular, we prove th...

The common structure of the space of pure states P of a classical and a quantum mechanical system is that of a Poisson space with a transition probability. This is a uniform space equipped with a Poisson structure, as well as with a function p : P × P → [0, 1], with certain properties. The Poisson structure is connected with the transition probabilities through unitarity: each point ρ ∈ P defin...

An admissible Poisson algebra (or briefly, an adm-Poisson algebra) gives equivalent presentation with only one operation for a algebra. We establish bialgebra theory algebras independently and systematically, including but beyond the corresponding results on bialgebras given in [27]. Explicitly, we introduce notion of which are to Manin triples as well bialgebras. The direct correspondence betw...

For a Banach algebra $fA$, we introduce ~$c.c(fA)$, the set of all $phiin fA^*$ such that $theta_phi:fAto fA^*$ is a completely continuous operator, where $theta_phi$ is defined by $theta_phi(a)=acdotphi$~~ for all $ain fA$. We call $fA$, a completely continuous Banach algebra if $c.c(fA)=fA^*$. We give some examples of completely continuous Banach algebras and a suffici...

We formulate Yang–Mills theory in terms of the large-N limit, viewed as a classical limit, of gauge-invariant dynamical variables, which are closely related to Wilson loops. We obtain a Poisson algebra of these dynamical variables corresponding to normal-ordered quantum (at a finite value of h̄) operators. Comparing with a Poisson algebra one of us introduced in the past for Weyl-ordered quantum...

A compact semisimple Lie algebra g induces a Poisson structure πS on the unit sphere S(g ) in g. We compute the moduli space of Poisson structures on S(g) around πS. This is the first explicit computation of a Poisson moduli space in dimension greater or equal than three around a degenerate (i.e. not symplectic) Poisson structure.

The elliptic algebra Aq,p(ŝl(N)c) at the critical level c = −N has an extended center containing trace-like operators t(z). Families of Poisson structures, defining q-deformations of the WN algebra, are constructed. The operators t(z) also close an exchange algebra when (−p1/2)NM = q−c−N for M ∈ Z. It becomes Abelian when in addition p = qNh where h is a non-zero integer. The Poisson structures...

In this article, we address one of the questions raised by Marc Rieffel in his collection of questions on deformation quantization. The question is whether the K-groups remain the same under flabby strict deformation quantizations. By “deforming” the question slightly, we produce a negative answer to the question. In his collection of questions on deformation quantization [8], Marc Rieffel aske...

A strict quantization of a Poisson manifold P on a subset I ⊆ R containing 0 as an accumulation point is defined as a continuous field of C∗-algebras {Ah̄}h̄∈I , with A0 = C0(P ), a dense subalgebra Ã0 of C0(P ) on which the Poisson bracket is defined, and a set of continuous cross-sections {Q(f )} f∈Ã0 for which Q0(f ) = f . Here Qh̄(f ∗) = Qh̄(f )∗ for all h̄ ∈ I , whereas for h̄ → 0 one requires t...