PT-symmetric non-commutative spaces with minimal volume uncertainty relations

We provide a systematic procedure to relate a three-dimensional q-deformed oscillator algebra to the corresponding algebra satisfied by canonical variables describing non-commutative spaces. The large number of possible free parameters in these calculations is reduced to a manageable amount by imposing various different versions of PT-symmetry on the underlying spaces, which are dictated by the specific physical problem under consideration. The representations for the corresponding operators are in general non-Hermitian with regard to standard inner products and obey algebras whose uncertainty relations lead to minimal length, areas or volumes in phase space. We analyze in particular one three-dimensional solution which may be decomposed into a two-dimensional non-commutative space plus one commuting space component, and also into a one-dimensional non-commutative space plus two commuting space components. We study some explicit models on these types of non-commutative spaces.