the multiplication map. We have $
m (\epsilon \otimes 1)\Delta = \imath : {\U^{\imath}}_q(\mathfrak{sl}_{2r+2}) \longrightarrow \U_q(\mathfrak{sl}_{2r+2})
$ by direct computation.

The $\imath$-canonical bases
----------------------------

The rest of the section we shall develop the theory of $\imath$-canonical bases for the quantum symmetric pairs $(\U_q(\mathfrak{sl}_{2r+1}), {\U^{\imath}}_q(\mathfrak{sl}_{2r+1}))$, $(\U_q(\mathfrak{sl}_{2r+2}), {\U^{\imath}}_q(\mathfrak{sl}_{2r+2}))$ and its applications. The formulation of the theory is uniform for both cases. Hence we shall drop the subscript, and denote both quantum symmetric pairs simply by $(\U, {\U^{\imath}})$, and denote the correspond index sets simply by $\I = \I_k$ and $\I^{\imath} = \I^{\imath}_k$, for $k= 2r+2$ or $2r+1$. In this section we shall assume all modules are finite dimensional.

Let $\widehat{\U}$ be the completion of the $\Qq$-vector space $\U$ with respect to the following descending sequence of subspaces $\U^+ \U^0 \big(\sum_{\hgt(\mu) \geq N}\U_{-\mu}^- \big)$, for $N \ge 1.$ Then we have the obvious embedding of $\U$ into $\widehat{\U}$. We let $\widehat{\U}^-$ be the closure of $\U^-$ in $\widehat{\U}$, and so $\widehat{\U}^- \subseteq \widehat{\U}$. By continuity the $\Q(q)$-algebra structure on $\U$ extends to a $\Q(q)$-algebra structure on $ \widehat{\U}$. The bar involution $\bar{\ }$ on $\U$ extends by continuity to an anti-linear involution on $\widehat{\U}$, also denoted by $\bar{\, }$. The following proposition is 