Type I factor representations and type I groups

Assume the Axiom of Choice. Let $H$ be a nonzero separable complex Hilbert space and let $M\subseteq \mathcal{B}(H)$ be a concrete factor von Neumann algebra (Factor (primary) representations). A nonzero projection $p\in M$ is minimal, or abelian, when $pMp=\mathbb{C}p$. The factor $M$ is of type I when it contains a nonzero minimal projection. A strongly continuous unitary representation $\pi$ of a topological group $G$ on a nonzero separable Hilbert space is a type I factor representation when its generated von Neumann algebra $\pi (G{)}^{\mathrm{\prime }\mathrm{\prime }}$ is a factor of type I; and a factor representation is of type I when it is a multiple of an irreducible representation (equivalently, by A separable type I factor is a multiple of an irreducible representation ↗, when $\pi (G{)}^{\mathrm{\prime }\mathrm{\prime }}$ contains a nonzero minimal projection). The group $G$ is type I when every factor representation of $G$ on a separable Hilbert space is of type I. The two descriptions of a type I factor representation agree: for a strongly continuous unitary representation $\pi$ on nonzero separable $H$ with $M=\pi (G{)}^{\mathrm{\prime }\mathrm{\prime }}$ a factor, $M$ contains a nonzero minimal projection if and only if there is an irreducible representation $\sigma$ of $G$ and $m\in \{1,2,\dots ,\mathrm{\infty }\}$ with $\pi \cong {\sigma }^{\oplus m}$ (A separable type I factor is a multiple of an irreducible representation ↗).

• The factor-to-multiple equivalence is proved locally in A separable type I factor is a multiple of an irreducible representation ↗. Its justified_by edge is a well-definedness discharge rather than a reverse logical prerequisite; the lemma depends on this Definition only for the minimal-projection/type-I terminology. Minimal projections in $M$ yield the multiplicity space, while minimal projections in $M^{\mathrm{\prime }}$ yield invariant irreducible carriers.

• Bekka Proposition 6.B.14 states the equivalence and gives a proof through earlier propositions, but that citation does not replace the required local supplier argument.

• For second-countable locally compact type-I groups, the precise all-separable-representation consequence is the canonical irreducible direct-integral decomposition and its measure-class/multiplicity uniqueness in Irreducible direct integral decomposition for type I groups and Essential uniqueness of the type I irreducible disintegration, obtained from central type-I factor fibres. This does not assert that every nonfactor generated von Neumann algebra is a factor of type I; the group terminology above tests factor representations.