# Existence of a weak-star limit of a family of operators in $B(\mathcal{H})$

I am working with a family of operators $$(A_{s,\alpha})_{s\geq 0,\alpha>0}$$ in the space $$B(\mathcal{H})$$ (the space of bounded linear operators on a complex separable Hilbert space $$\mathcal{H}$$) depending on two parameters $$s\geq 0$$ and $$\alpha>0$$. This family is such that for fixed $$\alpha>0$$, the mapping $$[0,\infty)\ni s\mapsto A_{s,\alpha}\in B(\mathcal{H})$$ is a strongly continuous semigroup , and $$(\star)\quad \forall\, s>0,\,\forall\, \alpha>0:\quad \big|\big|A_{s,\alpha}\big|\big|_{B(\mathcal{H}}\leq 1\,,$$ i.e. the family of operators $$(A_{s,\alpha})_{s\geq 0,\alpha>0}$$ lives in the norm-closed unit ball of $$B(\mathcal{H})$$. Next, the pre-dual of $$B(\mathcal{H})$$ (up to isometric isomorphism) is $$B_1(\mathcal{H})$$, the trace-class operators on $$\mathcal{H}$$. This defines a weak-star topology on $$B(\mathcal{H})$$, and by the Banach-Alaoglu theorem, the norm-closed unit ball of $$B(\mathcal{H})$$ is weak-star compact.
Question 1: I think that since $$\mathcal{H}$$ is separable, it follows that $$B_1(\mathcal{H})$$ is separable, and hence the norm-closed unit ball of $$B(\mathcal{H})$$ is weak-star sequentially compact?
Next, assuming an affirmative answer to Question 1, let us fix an arbitrary sequence $$\alpha_n\to 0^+$$. Then for any fixed $$s\geq 0$$, $$(\star)$$ above implies the sequence of operators $$(A_{s,\alpha_n})_{n\in \mathbb{N}}$$ has a weak-star convergent subsequence $$(A_{s,\alpha_{n_k}})_{k\in \mathbb{N}}$$. Unfortunately, however, this subsequence may depend on the particular $$s\geq 0$$.
Question 2: Is it possible to show that there exists a subsequence $$(\alpha_{n_j})_{j\in \mathbb{N}}$$ of the above $$\alpha_n\to 0^+$$ so that for all $$s\geq 0$$ the sequence of operators $$(A_{s,\alpha_{n_j}})_{j\in \mathbb{N}}$$ converges in the weak-star topology in $$B(\mathcal{H})$$? If not, are there some "natural" assumptions one can make on the $$\alpha$$-dependence of the operators such that this works?