By I. Kaplansky

An algebraic prelude Continuity of automorphisms and derivations $C^*$-algebra axiomatics and simple effects Derivations of $C^*$-algebras Homogeneous $C^*$-algebras CCR-algebras $W^*$ and $AW^*$-algebras Miscellany Mappings maintaining invertible components Nonassociativity Bibliography

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If G is a group, then the group algebra FG is a Hopf algebra, with antipode given by S(g) = g −1 , for all g ∈ G. Recall that if U and V are comodules over some bialgebra B, then their tensor product W = U ⊗ V is itself a comodule. A G-graded space A with multiplication μ : A ⊗ A → A is a G-graded algebra if and only if the following diagram commutes: A⊗A ρA ⊗ A μ ⊗ id μ A / A ⊗ A ⊗ FG ρA / A ⊗ FG This can be expressed by saying that ρA is a homomorphism of algebras or, equivalently, that μ is a homomorphism of comodules.

Reﬁnements and coarsenings. If we apply an arbitrary homomorphism α : G → H to a G-grading Γ, then some components of Γ may coalesce in α Γ. 24. Let Γ and Γ be two gradings on V with supports S and T , respectively. We will say that Γ is a reﬁnement of Γ , or that Γ is a coarsening of Γ, and write Γ ≤ Γ, if for any s ∈ S there exists t ∈ T such that Vs ⊂ Vt . If, for some s ∈ S, this inclusion is strict, then we will speak of a proper reﬁnement or coarsening. Clearly, ≤ is a partial order on the set of all gradings on V (if we regard all relabelings as one grading).

Now pick a homogeneous x ∈ I such that Ix = 0. By the minimality of I, we have Ix = I and annI (x) = 0, where annI (x) := {r ∈ I | rx = 0}. Hence there exists ε ∈ I such that εx = x. Replacing ε by its homogeneous component in Re , we may assume that ε has degree e. Since ε2 − ε ∈ annI (x), we conclude that ε2 = ε. Since Rε = 0, we have Rε = I by minimality. Let V be any graded simple left R-module. Since IV is a graded submodule of V , we have either IV = 0 or IV = V . But the action of R on V is faithful, so IV = V .