1204.1033 (Stephen Bruce Sontz)

Paragrassmann Algebras are Quantum Spaces having Reproducing Kernels and
Toeplitz Operators    [PDF]

Stephen Bruce Sontz

Paragrassmann algebras are given a sesquilinear form for which one subalgebra becomes a Hilbert space known as the Segal-Bargmann space. This Hilbert space and the ambient space of the paragrassmann algebra itself are shown to have reproducing kernels. These algebras are not isomorphic to algebras of functions so some care must be taken in defining what "evaluation at a point" corresponds to in this context. The reproducing kernel in the Segal-Bargmann space is shown to have most, though not all, of the standard properties. Toeplitz operators are then defined using the reproducing kernel in the usual way, and some of their properties are proved. These quantum spaces provide non-trivial examples of spaces which have a reproducing kernel but are not spaces of functions.

View original: http://arxiv.org/abs/1204.1033