Preservation of properties in nonstandard hulls of $C^*$--algebras

Abstract

IWe extend the nonstandard hull construction of Luxemburg and we show that the proposed extension covers various other constructions such as the tracial and the weak nonstandard hull. We provide a novel proof that the tracial nonstandard hull of an internal von Neumann algebra is itself a von Neumann algebra. We prove that some extensively studied properties of $C^*$--algebras are preserved and reflected by the nonstandard hull construction. Eventually, we show how to obtain hull--like representations of the two $K$--groups associated to the nonstandard hull of an internal $C^*$--algebra.