Coherent States on Hilbert Modules

TL;DR

This paper extends the concept of coherent states from Hilbert spaces to Hilbert modules over $C^*$-algebras, establishing their properties, relations to classical objects, and applications to positive kernels and dilations.

Contribution

It introduces a framework for coherent states on Hilbert modules over $C^*$-algebras, generalizing classical definitions and exploring their structural and functional properties.

Findings

Coherent states are well-defined on Hilbert modules with a compatible $C^*$-algebra action.

Classical objects like the Cuntz algebra relate to specific coherent states examples.

Coherent states induce completely positive kernels and enable dilation results for positive operator valued measures.

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, $\mathcal A$. 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 $C^*$-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.