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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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References:

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