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.
Thanks in advance for your help.
Best Answer
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].