Before I state my question, let me provide the definition of a compact quantum group.

Definition:An ordered pair $ \mathscr{G} = (\mathscr{A},\Phi) $ is called acompact quantum groupif

- $ \mathscr{A} $ is a separable and unital C*-algebra (whose identity element we denote by $ 1_{\mathscr{A}} $);
- $ \Phi $ is a unital $ * $-homomorphism from $ \mathscr{A} $ to $ \mathscr{A} \otimes_{\sigma} \mathscr{A} $, where ‘$ \sigma $’ means ‘minimal’;
- the diagram $$ \require{AMScd} \begin{CD} \mathscr{A} @>{\Phi}>> \mathscr{A} \otimes_{\sigma} \mathscr{A} \\ @V{\Phi}VV @VV{\Phi \otimes \text{id}}V \\ \mathscr{A} \otimes_{\sigma} \mathscr{A} @>{\text{id} \otimes \Phi}>> \mathscr{A} \otimes_{\sigma} \mathscr{A} \otimes_{\sigma} \mathscr{A} \end{CD} $$ commutes;
- the sets $$ \{ (a \otimes 1_{\mathscr{A}}) \Phi(b) \mid a,b \in \mathscr{A} \} \quad \text{and} \quad \{ (1_{\mathscr{A}} \otimes a) \Phi(b) \mid a,b \in \mathscr{A} \} $$ are linearly dense in $ \mathscr{A} \otimes_{\sigma} \mathscr{A} $.

In this set of notes on compact quantum groups by S. Woronowicz, one finds the following remarkable theorem whose proof relies upon nothing more than the representation theory of C*-algebras.

Theorem:Let $ \mathscr{G} = (\mathscr{A},\Phi) $ be a compact quantum group. Then there exists a unique state (i.e., a normalized positive linear functional) $ h $ on $ \mathscr{A} $ such that $$ h \star a = a \star h = h(a) \cdot 1_{\mathscr{A}}, $$ where $$ h \star a \stackrel{\text{def}}{=} (\text{id} \otimes h)(\Phi(a)) \quad \text{and} \quad a \star h \stackrel{\text{def}}{=} (h \otimes \text{id})(\Phi(a)). $$ We call $ h $the Haar state on $ \mathscr{G} $.

Out of curiosity, I tried to see how Woronowicz’s Theorem implies the existence of the classical Haar measure on a compact Hausdorff topological group $ G $. Despite my best efforts, I only managed to succeed in the case where $ G $ is additionally second-countable.

Let $ G $ be a compact Hausdorff topological group. Suppose further that it is second-countable (or equivalently, metrizable). Then $ C(G) $ ― the space of complex-valued continuous functions on $ G $ equipped with the supremum norm ― is a separable and unital C*-algebra.

Consequently, $ (C(G),\Delta) $ is a compact quantum group, where the co-multiplication $$ \Delta: C(G) \to C(G) ~ \widehat{\otimes} ~ C(G) \cong C(G \times G) $$ is defined by $$ \forall x,y \in G: \quad {\Delta f}(x,y) \stackrel{\text{def}}{=} f(x y). $$ According to Woronowicz’s Theorem, there exists a unique state $ h $ on $ C(G) $ such that $$ (\spadesuit) \quad h \star f = f \star h = h(f) \cdot \mathbf{1}. $$ View $ \Delta f $ as the limit of a net $ \displaystyle \left( \sum_{i \in I_{\lambda}} p_{\lambda,i} \otimes q_{\lambda,i} \right)_{\lambda \in \Lambda} $ of finite sums of elementary tensors. Then \begin{align*} \forall x \in G: \quad (h \star f)(x) & = [(\text{id} \otimes h)(\Delta f)](x) \\ & = \left[ (\text{id} \otimes h) \left( \lim_{\lambda} \sum_{i \in I_{\lambda}} p_{\lambda,i} \otimes q_{\lambda,i} \right) \right](x) \\ & = \left[ \lim_{\lambda} \sum_{i \in I_{\lambda}} p_{\lambda,i} \cdot h \left( q_{\lambda,i} \right) \right](x) \quad (\text{By continuity.}) \\ & = \lim_{\lambda} \sum_{i \in I_{\lambda}} {p_{\lambda,i}}(x) \cdot h \left( q_{\lambda,i} \right) \quad (\text{By continuity.}) \\ & = \lim_{\lambda} h \left( \sum_{i \in I_{\lambda}} {p_{\lambda,i}}(x) \cdot q_{\lambda,i} \right) \quad (\text{By the linearity of $ h $.}) \\ & = h \left( \lim_{\lambda} \sum_{i \in I_{\lambda}} {p_{\lambda,i}}(x) \cdot q_{\lambda,i} \right) \quad (\text{By continuity.}) \\ & = h({\Delta f}(x,\cdot)) \\ & = h({L_{x}}(f)), \end{align*} where $ L_{x}: C(G) \to C(G) $ denotes left translation by $ x $. Similarly, $$ \forall x \in G: \quad (f \star h)(x) = h({R_{x}}(f)), $$ where $ R_{x}: C(G) \to C(G) $ denotes right translation by $ x $. By $ (\spadesuit) $, $$ \forall x \in G: \quad h({L_{x}}(f)) = h({R_{x}}(f)) = h(f). $$ This implies that $ h $ is a left- and right-invariant state on $ C(G) $. It follows from the Riesz Representation Theorem that $ h $ corresponds to a unique regular Borel probability measure on $ G $ that is left- and right-invariant. This proves the existence of the classical Haar measure on $ G $.

My question is:

How can I strengthen the argument above so that it yields the existence of the classical Haar measure on a

generalcompact Hausdorff topological group? Is there a way to circumvent the requirement of separability in the definition of a compact quantum group?

Thank you very much for your help!