Math 2.0

Topological groups, Lie groups Definition

Type I group

Builds on Locally compact groupUnitary representationVon Neumann algebraCommutantIrreducible representationHilbert space

Definition

Let GG be a locally compact group, and let π\pi be a unitary representation of GG on a Hilbert space ℋ\mathcal{H}. Let π(G)′′\pi(G)'' be the von Neumann algebra generated by the operators π(g)\pi(g), the double commutant of {π(g):g∈G}\set{\pi(g) : g \in G}. The representation π\pi is a factor representation if the center of π(G)′′\pi(G)'' consists of the scalar multiples of the identity. The group GG is of type I if, for every factor representation π\pi, the algebra π(G)′′\pi(G)'' is a type I factor. This means that π(G)′′\pi(G)'' is isomorphic to the algebra B(𝒦)B(\mathcal{K}) of all bounded operators on some Hilbert space 𝒦\mathcal{K}.

Remarks

Equivalently, every factor representation of GG is a multiple of one irreducible representation. For second countable GG, Glimm's theorem gives further equivalent conditions; one of them is that two irreducible representations with the same kernel in C∗(G)C^*(G) are unitarily equivalent.

Example: every abelian locally compact group is a type I group. If GG is abelian, π(G)′′\pi(G)'' is commutative, so a factor representation has π(G)′′=ℂI\pi(G)'' = \C I. Hence each factor representation is a multiple of a one-dimensional representation, a character, and π(G)′′=ℂI≅B(ℂ)\pi(G)'' = \C I \cong B(\C). Compact groups, connected real semisimple Lie groups and connected nilpotent Lie groups are also type I groups.

Non-example: the free group F2F_2 on two generators is not a type I group. Its left regular representation generates a factor of type II1\mathrm{II}_1, which is not a type I factor. More generally, Thoma proved that a discrete group is of type I if and only if it has an abelian normal subgroup of finite index.

For type I unimodular groups there is a Plancherel theorem that decomposes the left regular representation over the unitary dual.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Commutant, concept, no evidence levelCommutantconceptHilbert space, concept, no evidence levelHilbert spaceconceptIrreducible representation, concept, no evidence levelIrreducible represent…conceptLocally compact group, concept, no evidence levelLocally compact groupconceptUnitary representation, concept, no evidence levelUnitary representationconceptVon Neumann algebra, concept, no evidence levelVon Neumann algebraconceptType I group, definition, no evidence levelType I groupdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Commutant concept
  2. Hilbert space concept
  3. Irreducible representation concept
  4. Locally compact group concept
  5. Unitary representation concept
  6. Von Neumann algebra concept
  7. Type I group definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Unitary_representation
About this page
TagK904QFC9BQ
URIm2://K904QFC9BQ
Kinddefinition
WritingW0 stub
MSC22D10, 22D25, 46L10
DefinesType I group
UsesLocally compact group, Unitary representation, Von Neumann algebra, Commutant, Irreducible representation, Hilbert space
Changed3 November 2026

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