## An invariant for finitely presented $$\mathbb{C} G$$-modules.(English)Zbl 0813.43004

For a group $$G$$, let $$\text{Map}(G)$$ be the category whose objects are $$\mathbb{C} G$$-module maps between finite-rank free $$\mathbb{C} G$$-modules and whose morphisms are pairs of intertwining module maps. A functor from $$\text{Map}(G)$$ to the category of representations of $$G$$ on Hilbert space and bounded intertwining maps is constructed such that the unitary $$G$$-space $${\mathcal H}(\Delta,G)$$ associated to an object $$\Delta$$ in $$\text{Map}(G)$$ depends only on the finitely presented $$\mathbb{C} G$$-module $$\text{coker }\Delta$$. The von Neumann dimension of $${\mathcal H}(\Delta,G)$$ gives a well-behaved numerical invariant for finitely presented modules (zero whenever $$G$$ is amenable, non-zero and readily calculable in certain other situations). In case $$G$$ is finitely generated and $$\text{coker }\Delta = \mathbb{C}$$, the representation of $$G$$ on $${\mathcal H}(\Delta, G)$$ has an interpretation in terms of actions of $$G$$ on directed graphs.

### MSC:

 43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis) 22D25 $$C^*$$-algebras and $$W^*$$-algebras in relation to group representations 20C07 Group rings of infinite groups and their modules (group-theoretic aspects) 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 43A07 Means on groups, semigroups, etc.; amenable groups 16S34 Group rings
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