There may well be an answer to this question in a simpler category than that of finite dimensional quantum groups and in that case this question is more suitable to math.stack and I apologise in advance if this is the case. Consider a finite dimensional quantum group $A=F(\mathbb{G})$ with a Haar state $h:A\rightarrow\mathbb{C}$. In the finite dimensional case (and in more generality in fact), this allows us to define '$p$-norms' on $A$ via $$\|a\|_p:=\sqrt[p]{h((a^*a)^{p/2})}.$$ In particular, $\|a\|_1=h((a^*a)^{1/2})$ and $\|a\|_2=\sqrt{h(a^*a)}$. There is a Cauchy-Schwarz Inequality: $$\|ab\|_1\leq\|a\|_2\|b\|_2.$$ We can define a dual quantum group $\hat{A}$ via the map $\mathcal{F}:A\mapsto A'$, $\mathcal{F}(a)(b)=h(ba)$. In this, finite dimensional case, $\hat{A}=A'$, and the multiplication in $\hat{A}$ is given by the convolution: $$\nu\star\mu=(\nu\otimes\mu)\Delta,$$ the Haar state $\hat{h}:\hat{A}\rightarrow\mathbb{C}$ is given by $$\hat{h}(\mathcal{F}(a))=\varepsilon(a),$$ and the involution is $$\nu^*(a)=\overline{\nu(S(a)^*)}.$$ I am interested in finding bounds for $$\|\nu\|_1=\hat{h}((\nu^*\nu)^{1/2}).$$ Via $\varepsilon\star \nu=\nu$, we have the following upper bound on $\|\nu\|_1$: $$\|\nu\|_1=\|\varepsilon\star \nu\|_1\leq\|\varepsilon\|_2\|\nu\|_2=\sqrt{\dim A}\cdot\|\nu\|_2.$$ For my application, I have a way of calculating and bounding $\|\nu\|_2$ above but I am also interested in bounding below: $$?\leq \|\nu\|_1\leq \sqrt{\dim A}\|\nu\|_2.$$ In the classical case where $G$ is a finite group, a 1-norm on $F(G)$ might be given by $$\|f\|_1'=\sum_{t\in G}|f(t)|,$$ and this fits quite well into this framework: $$\|f\|_1=h\left((f^*f)^{1/2}\right)=h(|f|)=\frac{1}{|G|}\sum_{t\in G}|f(t)|.$$ Now I am more interesting in looking at $\|\cdot\|_1:\hat{A}\rightarrow[0,\infty)$. Now classically one might define the 1-norm on $\mathbb{C}G\supset M_p(G)$ as $$\|\nu\|_1'=\sum_{t\in G}|\nu(\delta_t)|.$$ An advantage of working with this norm is that we have $$\|\nu\|_1'=\sup_{\underset{f\in F(G)}{\|f\|_{\infty}\leq1}}|\nu(f)|,$$ so that we can generate lower bounds by looking at test functions $\phi\in F(G)^1$ and so we have $$|\nu(\phi)|\leq \|\nu\|_1'\leq \sqrt{|G|}\|\nu\|_2'.$$ The problem with using $\|\cdot\|_1:\hat{A}\rightarrow[0,\infty)$ via the Haar state $\hat{h}$: $$\|\nu\|_1=\hat{h}((\nu^*\nu)^{1/2}),$$ is that even in the classical case I don't quite have something like $$\|\nu\|_1=\frac{1}{|G|}\sum_{t\in G}|\nu(\delta_t)|.$$ When we are in $A=F(G)$ the involution is simply $$f^*(s)=\overline{f(s)},$$ and with pointwise multiplication and positivity in the C*-algebra equivalent to positivity of the coefficients, we have $$(f^*f)^{1/2}(s)=|f(s)|,$$ so the 1-norm works quite nicely in there. Things are more complicated however in $\hat{A}=\mathbb{C}G$ (as is alluded to in [this question of mine][1]). In general, even for symmetric probability measures, we don't have $$(\nu^*\nu)^{1/2}=\nu.$$ Now in the classical case I can just use $\|\cdot\|_1'$ and the ordinary C-S to get my upper bounds. However in the truly non-commutative case I want to use $\|\cdot\|_1$... if I can get myself some lower bounds! Otherwise I can just use and bound $\|\cdot \|_2$ above. > In the not-necessarily-commutative, quantum group case, is there a way to generate lower bounds on $\|\nu\|_1=\hat{h}((\nu^*\nu)^{1/2})$ via 'test functions': $$\|\nu\|_1\geq\sup_{s\in S}F(s,\nu)?$$ Perhaps $F(s,\nu)$ involves convolving $\nu$ with some element of $s\in S\subset\hat{A}$ with a small norm or maybe something like hitting $\nu$ with an element of $s\in S\subset A$ with small norm: $\nu(s)$. Thank you for your help. [1]: https://math.stackexchange.com/questions/1320060/positive-elements-of-mathbbcg-as-functionals-versus-as-elements-of-the-c