Exercise 26, § 11

Let C be a coassociative and counital cogebra over a commutative field K.

a) For every vector subspace E of C which is a $C^*$-left module for the law $(u, x) \mapsto u \cdot x$, the left annihilator of this $C^*$-module is a two-sided ideal $\mathfrak{J}'$ of the algebra $C^*$. Prove that if E is of finite dimension over K, $C^*/\mathfrak{J}'$ is of finite dimension.

b) Deduce from a) that for every element $x \in C$, the subcogebra of C generated by x is of finite dimension. (Note that the $C^*$-module E generated by x is of finite dimension, and that if $\mathfrak{J}'$ is its left annihilator, one has $x \in \mathfrak{{J}'}^0$, orthogonal of $\mathfrak{J}'$ in C.)