A Banach algebra encoding quantum group duality
We introduce and study a new Banach algebra structure on the trace-zero subspace $\mathcal{T}(L^2(\mathbb{G}))_0$ of trace class operators for any locally compact quantum group $\mathbb{G}$; it is defined through a mixed Lie-type product of the two dual products on $\mathcal{T}(L^2(\mathbb{G}))$ arising from the canonical extensions of the co-products of $\mathbb{G}$ and $\widehat{\mathbb{G}}$. The surprising fact that this new product is indeed associative stems precisely from the duality of the latter two products. This, in particular, gives new faithful associative products on trace-zero matrices in $M_d(\mathbb{C})$. After establishing some basic properties, we show that the single algebra $\mathcal{T}(L^2(\mathbb{G}))_0$ captures simultaneous properties of $\mathbb{G}$ and $\widehat{\mathbb{G}}$, is faithful for a large class of quantum groups, and encodes both $M^r_{cb}(L^1(\mathbb{G}))$ and $M^r_{cb}(L^1(\widehat{\mathbb{G}}))$ as left, respectively right, completely bounded module maps on $\mathcal{T}(L^2(\mathbb{G}))$. We finish by exhibiting an analogous product on the trace-zero nuclear operators $\mathcal{N}(L^p(G))_0$ for a locally compact group $G$ and $p\in(1,\infty)$. Building on [7], our work suggests an approach for developing an $L^p$-version of locally compact quantum group theory.