# Reference request: decomposability of $\mathbb{G}$-Hilbert modules

Let $$\mathbb{G}$$ be a compact quantum group, $$B$$ be a $$C^*$$-algebra together with a right action $$\beta: B \to B\otimes C(\mathbb{G})$$ which is a non-degenerate $$*$$-homomorphism satisfying $$(\beta \otimes \iota)\beta = (\iota \otimes \Delta)\beta$$ and the Podles density condition. A right $$B$$-Hilbert module $$X$$ is called $$\mathbb{G}$$-equivariant if it is equipped with a coaction $$\delta: X \to X\otimes C(\mathbb{G})$$ such that

• $$\beta(\langle x,y\rangle_X) = \langle\delta(x), \delta(y)\rangle_{X\otimes C(\mathbb{G})}$$
• $$(\delta \otimes \iota)\delta = (\iota \otimes \Delta)\delta$$
• $$[\delta(X)(1\otimes C(\mathbb{G}))]= X\otimes C(\mathbb{G})$$.

Example: Let $$U\in B(H_U)\otimes C(\mathbb{G})$$ be a finite-dimensional representation of $$\mathbb{G}$$. Then the right $$B$$-Hilbert module $$H_U\otimes B$$ becomes $$\mathbb{G}$$-equivariant for $$\delta(\xi\otimes b) = U_{13} (\xi \otimes \beta(b)).$$ We denote this $$\mathbb{G}$$-equivariant module by $$B\times U.$$

I'm looking for references/proofs of the following facts:

(1) If $$X$$ is finitely generated as a right Hilbert module and $$\mathbb{G}$$-equivariant, then $$X$$ embeds equivariantly in a finite direct sum $$\bigoplus_{i=1}^n (B\times U_i)$$ where the $$U_i$$ are irreducible representations of $$\mathbb{G}$$.

(2) If $$\beta$$ is a homogeneous action, i.e. $$B$$ is unital and $$\{b\in B: \alpha(b) = b\otimes 1\}= \mathbb{C}1_B$$, then every $$\mathbb{G}$$-equivariant right Hilbert $$B$$-module decomposes as a direct sum of equivariant irreducible submodules.

Let's assume that $$G$$ is a reduced compact quantum group, that is, the Haar state on $$C(G)$$ is faithful.