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


* $\beta(\langle x,y\rangle\_X) = \langle\delta(x), \delta(y)\rangle\_{X\otimes C(\mathbb{G})}$
* $(\delta \otimes \iota)\delta = (\iota \otimes \Delta)\delta$
* $[\delta(X)(1\otimes C(\mathbb{G}))]= X\otimes C(\mathbb{G})$.




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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 $B\times U.$




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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.




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Thanks in advance for your help.



Reference request: decomposability of $\mathbb{G}$-Hilbert modules
Let's assume that $G$ is a reduced compact quantum group, that is, the Haar state on $C(G)$ is faithful.


(1): A direct reference is [1, Lemma 4.2]. You can also get this from a careful study of [2, Théorème 3.2].


(2): This (for countably generated modules) follows from [2, Théorème 3.2] and [3, Proposition 4.6].


1. *Neshveyev, Sergey; Tuset, Lars*, [**Hopf algebra equivariant cyclic cohomology, (K)-theory and index formulas**](https://doi.org/10.1023/B:KTHE.0000031399.40342.7d), (K)-Theory 31, No. 4, 357-378 (2004). [ZBL1067.19002](https://zbmath.org/?q=an:1067.19002).
2. Roland Vergnioux, KK-théorie équivariante et opératour de Julg-Valette pour les groupes quantiques, Ph.D. thesis, Universit ́e Paris Diderot-Paris 7, 2002.
3. *De Commer, Kenny; Yamashita, Makoto*, [**Tannaka-Krein duality for compact quantum homogeneous spaces. I: General theory**](https://www.emis.de/journals/TAC/volumes/28/31/28-31abs.html), Theory Appl. Categ. 28, 1099-1138 (2013). [ZBL1337.46045](https://zbmath.org/?q=an:1337.46045).


