Is unitary group of a von Neumann algebra is locally compact in some topology? Can we make sense of integration with respect to Haar measure?
2026-03-25 19:05:17.1774465517
About unitary group of a von Neumann algebra
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The unitary group is compact for finite von Neumann algebras of type I. It is also locally compact for the discrete topology.
Other than these two cases, the set of unitary elements can be equipped with the ultraweak topology, which yields a compact topolpogical space but for which the operation of multiplication is not continuous, or it can be equipped with the norm or ultrastrong topologies, which are not locally compact, but do give a topological group. In either case, the Haar theorem is not applicable.