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 $C^{\ast }$-algebras are the natural settings for a generalization of coherent states defined on Hilbert spaces. We consider those Hilbert $C^{\ast }$-modules which have a natural left action from another $C^{\ast }$-algebra say, $\mathcal{A}$. The coherent states are well defined in this case and they behave well with respect to the left action by $\mathcal{A}$. 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 $C^{\ast }$-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.

References in corpus (2)

• Formulation, Interpretation and Application of non-Commutative Quantum Mechanics
• Noncommutative quantum mechanics -- a perspective on structure and spatial extent

Cited by in corpus (2)

• Regular subspaces of a quaternionic Hilbert space from quaternionic Hermite polynomials and associated coherent states
• $\mathcal{D}$-Pseudo-Bosons, Complex Hermite Polynomials, and Integral Quantization