  We initiate a mathematically rigorous study of Klein-Gordon position
operators in single-particle relativistic quantum mechanics. Although not
self-adjoint, these operators have real spectrum and enjoy a limited form of
spectral decomposition. The associated C*-algebras are identifiable as crossed
products. We also introduce a variety of non self-adjoint operator algebras
associated with the Klein-Gordon position representation; these algebras are
commutative and continuously (but not homeomorphically) embeddable in
corresponding function algebras. Several open problems are indicated.
