Representations and irreducibility

A closed subspace K is invariant if every represented algebra element sends K into K. Because the representation contains adjoints, its orthogonal complement is invariant too: testing an image of a vector in K-perpendicular against a vector in K moves the operator to its adjoint. Such a subspace is reducing. Its orthogonal projection commutes with every represented operator. Conversely, the range of a projection in the commutant is reducing, so the two descriptions are equivalent.

The commutant consists of all bounded operators commuting with the represented algebra. A representation is irreducible if its only closed invariant subspaces are zero and the entire space. For a complex *-representation, the commutant is then exactly the scalar multiples of the identity, and conversely. This version of Schur's lemma uses bounded operators and complex scalars; it is not the same assertion for arbitrary real representations or for unbounded operators without domain control.

Faithful means the representation has zero kernel: no nonzero algebra element is represented by the zero operator. Cyclic means some vector has a dense algebra orbit. A faithful representation can be reducible, as coordinatewise multiplication of C² on C² shows. That representation is also cyclic, because multiplying the vector (1,1) produces every coordinate pair. Its two coordinate axes nevertheless remain invariant. Thus even the combination of faithfulness and cyclicity does not force irreducibility.

The standard action of M₂(C) on C² is faithful and irreducible, and every nonzero vector is cyclic. In contrast, an irreducible representation of a direct-sum algebra may kill an entire summand and therefore be nonfaithful. For example, acting with only the first M₂ summand of M₂⊕M₂ on C² gives an irreducible representation with nonzero kernel. These examples keep an algebraic property of the map separate from a geometric property of the action.

A state is pure when it is an extreme point of the convex state space: it cannot be written as a nontrivial convex mixture of two distinct states. The GNS representation of a state is irreducible exactly when that state is pure. This is a structural theorem, stated here without its full functional-analytic proof. It explains why commutative pure states are point evaluations, while pure matrix states correspond to rank-one density matrices. Purity refers to the whole observable algebra being considered.

Restrict a pure matrix state to the diagonal subalgebra and it may become a nontrivial classical mixture. For the unit vector (1,1)/√2, the full M₂ vector state is pure, but diagonal observables are averaged with equal weights on their two coordinates. Off-diagonal observables detect coherence that the diagonal subalgebra cannot see. The state's faithfulness is another issue: a rank-one state on M₂ annihilates a nonzero positive projection even though its GNS representation is the faithful irreducible standard action.