On the Shilov boundary ideal for Fréchet local operator systems
We show that the Shilov boundary ideal for a separable Fréchet local operator system is given by the intersection of the kernels of all its $Γ$-boundary representations.
math.OA↗
arXiv subjects
Gheorghe-Ionuţ Şimon
Publications and source records attributed to Gheorghe-Ionuţ Şimon.
Injective envelopes for locally C*-algebras
We introduce the notion of admissible injective envelope for a locally C*-algebra and show that each object in the category whose objects are unital Fréchet locally C*-algebras and whose morphisms are unital admissible local completely positive maps has a unique admissible injective envelope. The concept of admissible injectivity is stronger than that of injectivity. As a consequence, we show that a unital Fréchet locally W*-algebras is injective if and only if the C*-algebras from its Arens-Michael decomposition are injective.
math.OA↗
The R-Shilov boundary for a local operator space
To extend the notion of the injective envelope of a unital operator space to the locally convex case, Dosi (2014) first introduced the notion of the injective R-envelope for a unital operator space and then defined the injective R-envelope for a unital local operator space as the closure of the injective R-envelope for its bounded part. In this paper, we investigate the existence of the Shilov boundary ideal in this context, as defined by Arveson (1969). To do this, by following the conceptual frameworks underlying Hamana's constructions of the injective envelope and the C*-envelope, respectively, we define the notions of the injective R-envelope and the R-C*-envelope for a unital local operator space. Furthermore, we show that the injective R-envelope construction given by us coincides with the one given by Dosi (2014).
math.OA↗