Single sided multiplier Hopf algebras

Let $A$ be a non-degenerate algebra over the complex numbers and $\Delta$ a homomorphism from $A$ to the multiplier algebra $M(A\otimes A)$. Consider the linear maps $T_1$ and $T_2$ from $A\otimes A$ to $M(A\otimes A)$ defined by \begin{equation*} T_1(a\otimes b)=\Delta(a)(1\otimes b) \qquad\text{and}\qquad T_2(c\otimes a)=(c\otimes 1)\Delta(a). \end{equation*} The pair $(A,\Delta)$ is a multiplier Hopf algebra if these two maps have range in $A\otimes A$ and are bijections from $A\otimes A$ to itself. In our recent paper on the Larson-Sweedler theorem, single sided multiplier Hopf algebras emerge in a natural way. For this case, instead of requiring the above for the maps $T_1$ and $T_2$, we now have this property for the maps $T_1$ and $T_4$ or for $T_2$ and $T_3$ where \begin{equation*} T_3(a\otimes b)=(1\otimes b)\Delta(a) \qquad\text{and}\qquad T_4(c\otimes a)=\Delta(a)(c\otimes 1). \end{equation*} As it turns out, also for these single sided multiplier Hopf algebras, the existence of a unique counit and antipode can be proven. In fact, rather surprisingly, using the properties of the antipode, one can actually show that for a single sided multiplier Hopf algebra all four canonical maps are bijections from $A\otimes A$ to itself. In other words, $(A,\Delta)$ is automatically a regular multiplier Hopf algebra. We take the advantage of this approach to reconsider some of the known results for a regular multiplier Hopf algebra.