Coherent states on Hilbert modules

Abstract

We generalize the concept of coherent states, traditionally defined as special families of vectors on Hilbert spaces, to Hilbert modules. We show that Hilbert modules over C * -algebras are the natural settings for a generalization of coherent states defined on Hilbert spaces. We consider those Hilbert C * -modules which have a natural left action from another C * -algebra say, A. The coherent states are well defined in this case and they behave well with respect to the left action by A. Certain classical objects like the Cuntz algebra are related to specific examples of coherent states. Finally we show that coherent states on modules give rise to a completely positive kernel between two C * -algebras, in complete analogy to the Hilbert space situation. Related to this there is a dilation result for positive operator valued measures, in the sense of Naimark. A number of examples are worked out to illustrate the theory.

Regular subspaces of a quaternionic Hilbert space from quaternionic Hermite polynomials and associated coherent states

“…The fact that the analogues of the usual canonical coherent states cannot be built using a group theoretical argument in the case of quaternions, has been elaborated in [2]. On the other hand, analogues of such coherent states in a quaternionic setting have been constructed using other methods in [3] and [19]. In this paper we study the possibility of constructing some analogues of the so-called non-linear coherent states on quaternionic Hilbert spaces, using the recently developed holomorphic function theory for quaternionic variables [7,11,12].…”

General construction of reproducing kernels on a quaternionic Hilbert space

“…We end this section with a simple example of a reproducing kernel and its associated coherent states, which is the quaternionic equivalent of the canonical coherent states of physics (see, for example, [3]). These coherent states have also been reported in [4,33]. Consider the set of quaternionic monomials, (4.31) f n (q) = q n √ n!…”

D-Pseudo-Bosons, Complex Hermite Polynomials, and Integral Quantization

“…, ∞. These operators were introduced in [7], where they were used to construct coherent states on C * -Hilbert modules. The following properties are easily proved.…”