Type I locally compact group - Imported Knowlpedia Content

Definition

A locally compact group $G$ is a type I locally compact group if its full group $C^*$ -algebra $C^*(G)$ is a type I $C^*$ -algebra. Equivalently, every factor representation of $G$ generates a type I factor, or every factor representation is a Hilbert-space multiple of an irreducible representation Bekka–de la Harpe, §6.D. The definition itself does not require second countability, but separability or second-countability hypotheses are normally added when using measurable disintegration over the unitary dual.

Representation-theoretic consequence

For a second-countable type I group, the unitary dual has the standard Borel regularity needed for decomposition theory, and unitary representations admit essentially unique direct-integral decompositions into irreducibles with multiplicity data. Without the type I hypothesis, central decomposition into factor representations remains available in suitable separable settings, but those factors need not be irreducible multiples and decomposition into irreducibles need not give a workable measurable classification Folland, §7.4.

Examples and non-examples

Abelian and compact locally compact groups are type I. Connected nilpotent Lie groups and the standard classes of real reductive Lie groups are major noncompact examples. In contrast, the free group on two generators, with the discrete topology, is not type I; more generally, a countable discrete group is type I exactly when it is virtually abelian Bekka–de la Harpe, §8.B and Theorem 8.F.3. These examples show that amenability alone does not characterize the type I property.

Conventions and scope

References

Jacques Dixmier, $C^*$ -Algebras, North-Holland Mathematical Library 15, North-Holland, 1977. Publisher record. Relevant: §§9.1 and 13.9 on type I algebras and representations, and §18.8 on unitary representations and disintegration.
Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: §7.4 on type I groups and direct-integral decomposition.
Bachir Bekka and Pierre de la Harpe, Unitary Representations of Groups, Duals, and Characters, Mathematical Surveys and Monographs 250, American Mathematical Society, 2020. AMS record. Relevant: §6.D on type I representations, §8.B on type I groups, and Theorem 8.F.3 on discrete groups.