  Position deformation of a Heisenberg algebra and Hilbert space representation
of both maximal length and minimal momentum uncertainties may lead to loss of
Hermiticity of some operators that generate this algebra. Consequently, the
Hamiltonian operator constructed from these operators are also not Hermitian.
In the present paper, with an appropriate positive-definite Dyson map, we
establish the Hermiticity of these operators by means of a quasi-similarity
transformation. We then construct Hilbert space representations associated with
these quasi-Hermitian operators that generate a quasi-Hermitian Heisenberg
algebra. With the help of these representations we establish the path integral
formulation of any systems in this quasi-Hermitian algebra. Finally, using the
path integral of a free particle as an example, we demonstrate that the
Euclidean propagator, action, and kinetic energy of this system are constrained
by the standard classical mechanics limits.
