Earlier in his career Mackey did significant work in the
duality theory of
locally convex spaces, which provided tools for subsequent work in this area, including
Alexander Grothendieck's work on
topological tensor products. Mackey was one of the pioneer workers in the intersection of
quantum logic, the theory of
infinite-dimensional unitary representations of
groups, the theory of
operator algebras and
noncommutative geometry. A central role in Mackey's work, both in the theory of group representations and in
mathematical physics, was played by the concepts of
system of imprimitivity and
induced representations. This idea led naturally to an analysis of the representation theory of
semi-direct products in terms of ergodic actions of groups and in some cases a complete classification of such representations. Mackey's results were essential tools in the study of the representation theory of
nilpotent Lie groups using the
method of orbits developed by
Alexandre Kirillov in the 1960s. His notion of "virtual subgroup", introduced in 1966 using the language of
groupoids, had a significant influence in
ergodic theory. Another essential ingredient in Mackey's work was the assignment of a
Borel structure to the
dual object of a
locally compact group (specifically a locally compact separable metric group)
G. One of Mackey's important conjectures, which was eventually solved by work of
James Glimm on
C*-algebras, was that
G is
type I (meaning that all its factor representations are of type I) if and only if the Borel structure of its dual is a
standard Borel space. He has written numerous survey articles connecting his research interests with a large body of mathematics and physics, particularly
quantum mechanics and
statistical mechanics. ==Honours and students==