Simplicial and dimension groups with group action and their realization

Abstract

We define simplicial and dimension $\Gamma$-groups, the generalizations of simplicial and dimension groups to the case when these groups have an action of an arbitrary group $\Gamma.$ Assuming that the integral group ring of $\Gamma$ is Noetherian, we show that every dimension $\Gamma$-group is isomorphic to a direct limit of a directed system of simplicial $\Gamma$-groups and that the limit can be taken in the category of ordered groups with order-units or generating intervals. We adapt Hazrat's definition of the Grothendieck $\Gamma$-group $K_0^{\Gamma}(R)$ for a $\Gamma$-graded ring $R$ to the case when $\Gamma$ is not necessarily abelian. If $G$ is a pre-ordered abelian group with an action of $\Gamma$ which agrees with the pre-ordered structure, we say that $G$ is {\em realized} by a $\Gamma$-graded ring $R$ if $K_0^{\Gamma}(R)$ and $G$ are isomorphic as pre-ordered $\Gamma$-groups with an isomorphism which preserves order-units or generating intervals. We show that every simplicial $\Gamma$-group with an order-unit can be realized by a graded matricial ring over a $\Gamma$-graded division ring. If the integral group ring of $\Gamma$ is Noetherian, we realize a countable dimension $\Gamma$-group with an order-unit or a generating interval by a $\Gamma$-graded ultramatricial ring over a $\Gamma$-graded division ring. We also relate our results to graded rings with involution which give rise to Grothendieck $\Gamma$-groups with actions of both $\Gamma$ and $\mathbb Z_2$. We adapt the Realization Problem for von Neumann regular rings to graded rings and concepts from this work and discuss some other questions.