Factorial representation

Every irreducible representation is factorial because $\pi(A)'=\mathbb C I_H$ implies $\pi(A)''=\mathcal B(H)$. The converse fails: if a non-type-I factor $M$ acts by left multiplication on $L^2(M)$, then the generated algebra is a factor, but the commuting right action is nontrivial. Factoriality therefore captures indecomposability at the level of central summands, not the absence of all invariant subspaces Takesaki, Chapter V.

A central projection $z\in Z(\pi(A)'')$ splits $H$ into the reducing subspaces $zH$ and $(1-z)H$. Factoriality says that no such nontrivial split is available inside the generated von Neumann algebra. General representations can often be decomposed, in a measure-theoretic sense, into factorial representations; this is the factor analogue of decomposing a finite-dimensional representation into primary pieces.

A state on $A$ is called factorial when its GNS representation is factorial. Pure states yield irreducible, hence factorial, GNS representations, but factorial states need not be pure. This distinction is important for equilibrium states and for representations generating type II or type III factors.

1. Jacques Dixmier, C-Algebras*, North-Holland, 1977. Publisher record. Relevant: Chapter 5 on types of representations and primary representations.
2. Masamichi Takesaki, Theory of Operator Algebras I, Springer, 2002. Publisher record. Relevant: Chapter V on factors and factorial representations.