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Symmetric imprimitivity theorems for graph \(C^ *\)-algebras. (English) Zbl 1116.46305

Summary: The \(C^*\)-algebra \(C^*(E)\) of a directed graph \(E\) is generated by partial isometries satisfying relations which reflect the path structure of the graph. A. Kumjian and D. Pask [Ergodic Theory Dyn. Syst. 19, 1503–1519 (1999; Zbl 0949.46034)] considered the action of a group \(G\) on \(C^*(E)\) induced by an action of \(G\) on \(E\). They proved that if \(G\) acts freely and \(E\) is locally finite, then the crossed product \(C^*(E)\times G\) is Morita equivalent to the \(C^*\)-algebra of the quotient graph \(E/G\). This theorem bears a striking resemblance to a famous theorem of Green, which says that the crossed product \(C_0(X) \times G\) associated to a free and proper action of \(G\) on a locally compact space \(X\) is Morita equivalent to \(C_0(X/G)\). So one naturally asks whether this resemblance can be pushed further: are there analogues for free actions on graphs of the other Morita equivalences associated to free and proper actions on spaces? Here we contribute to this circle of ideas by proving an analogue of the symmetric imprimitivity theorem concerning commuting free and proper actions of two different groups.

MSC:

46L55 Noncommutative dynamical systems

Citations:

Zbl 0949.46034
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References:

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