Irreducible representation of a C*-algebra - Imported Knowlpedia Content

Definition

Let $\pi:A\to B(H)$ be a nonzero representation of a $C^*$ -algebra. It is irreducible when the only closed subspaces $K\subseteq H$ satisfying $\pi(a)K\subseteq K$ for every $a\in A$ are $K=\{0\}$ and $K=H$ . Because $A$ is closed under involution, every invariant closed subspace is automatically reducing: its orthogonal complement is invariant as well. The word “closed” is essential; irreducibility is a topological condition on the Hilbert-space representation, not algebraic simplicity of the underlying module.

Equivalent characterizations

Irreducibility is equivalent to the commutant condition

\pi(A)'=\mathbb C I_H.

It is also equivalent to every nonzero vector of $H$ being cyclic, since the closure of $\pi(A)\xi$ is an invariant subspace. These equivalences are standard forms of Schur's lemma for $C^*$ -representations Murphy, section 3.3.

Kernels and states

The kernel of an irreducible representation is a primitive ideal. Through the GNS construction, pure states yield irreducible representations, and every irreducible representation is equivalent to one obtained from a pure state after choosing a unit vector. Thus irreducibles connect state-space geometry with ideal structure.

Examples and non-examples

The defining representation of the compact-operator $C^*$ -algebra $K(H)$ on a nonzero Hilbert space $H$ is irreducible. A direct sum $\pi_1\oplus\pi_2$ with both summands nonzero is reducible because each summand is a proper closed invariant subspace. An irreducible representation may have a nonzero kernel, so it need not be faithful.

References

Gerald J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. Publisher record. Relevant: section 3.3 on irreducible and cyclic representations.
Jacques Dixmier, $C^*$ -Algebras, North-Holland, 1977. Publisher record. Relevant: section 2.5 on irreducible representations.