Let $A$ be a bialgebra. We can consider $A$ as a relfexive algebra (i.e. $A\cong A^{o*}$) or relfexive coalgebra (i.e. $A\cong A^{*o}$ where in each case $o$ denotes what is sometimes called restricted dual or simply dual coalgebra of an algebra). Equivalently, one can say that $A$ is reflexive as a coalgebra if and only if every cofinite ideal of $A^*$ is closed.

Concerning $A$ considered as a coalgebra, I found a comprehensive study by Heyneman-Radford online which sorts out the question when it is reflexive. It is called "Reflexivity and Coalgebras of Finite Type". It contains for example, as Theorem 4.2.6, the result that an almost connected coalgbera $C$ is reflexive if and only if $C_1$ is finite dimensional, where $C_0\subseteq C_1\subseteq\cdots$ is the coradical filtration of $C$.

My question concerns reflexivity of a bialgebra $A$ considered as both an algbera and coalgebra. When is a bialgebra reflexive considered as an algebra and how does this property relate to $A$ considered as relfexive as a coalgebra? Can we, somehow, connect the properties of $A$ being relfexive as an algebra and $A$ being reflexive as a coalgebra? Is there a similar study to Heyneman-Radford concerning reflexive algebras or bialgebras? An online search gives results about functional analysis which might talk about a different reflexivity notion.

Remark. If it makes a difference that an antipode is present, you can always assume that $A$ is actually a (possibly braided) Hopf algebra and treat the analogous question seperately for such a (possibly braided) Hopf algebra.