Virtual Groups 45 Years Later

In 1961 George Mackey introduced the concept of a virtual group as an equivalence class under similarity of ergodic measured groupoids, and he developed this circle of ideas in subsequent papers in 1963 and 1966. The goal here is first to explain these ideas, place them in a larger context, and then to show how they have influenced and helped to shape developments in four different but related areas of research over the following 45 years. These areas include first the general area and the connections between ergodic group actions, von Neumann algebras, measurable group theory and rigidity theorems. Then we turn to the second area concerning topological groupoids, C∗-algebras, K-theory and cyclic homology, or as it is now termed non-commutative geometry. We briefly discuss some aspects of Lie groupoids, and finally we shall turn attention to the fourth area of Borel equivalence relations seen as a part of descriptive set theory. In each case we trace the influence that Mackey’s ideas have had in shaping each of these four areas of research.