John Lindsay Orr

Let be a unital algebra and a proper left ideal. The quotient algebra is a vector space, and each acts on by left multiplication as . The map is a multiplicative, linear map from to the algebra of linear maps on and so is a representation of on , called the left regular representation.

Lemma 1. The left regular representation is irreducible if and only if is a maximal left ideal.

Proof. Suppose the left regular representation is irreducible and that is a left ideal. Then is clearly a subspace of and gives a subrepresentation, so that is either or . Thus is either or . Since these are the only two possibilities, must be a maximal left ideal.

Conversely, suppose is a maximal left ideal and suppose is a subrepresentation. Clearly is a left ideal conatining and so is either or . Thus is either or .

Definition 2. A primitive ideal is the kernel of an irreducible representation.

Remark 3. Given a proper left ideal , the kernel of the left regular representation is . For iff for all iff for all .