# Weak* separability of dual unit ball of D[0,1]

Let $$D[0,1]$$ be the space of all right-continuous left-limited functions $$f\colon [0,1]\to \mathbb{R}$$ equipped with the supremum norm $$f\mapsto \|f\|_\infty = \sup_{t\in[0,1]} |f(t)|$$. This is a non-separable Banach space whose dual $$D[0,1]^\ast$$ is known to be separable in the weak* topology; see, e.g., Chapter 41, p. 1756 of

Johnson, W. B. (ed.); Lindenstrauss, J. (ed.), Handbook of the geometry of Banach spaces. Volume 2, Amsterdam: North-Holland. xii, 1007-1866 (2003). ZBL1013.46001.

Is the unit ball in $$D[0,1]^\ast$$ separable in the weak* topology?

• Could you briefly describe the countable dense set?
– Ruy
May 19 at 15:34

$$(D([0,1]),\|\cdot\|_{\infty})$$ is a commutative $$C^{\ast}$$-algebra, so it is isometrically isomorphic to $$C(\Delta)$$ by the Gelfand map, where $$\Delta$$ is the character space of $$D([0,1])$$.
Let $$h_{1+}\in\Delta$$ be defined by $$h_{1+}(f) = f(1)$$ for all $$f\in (D([0,1])$$. Every $$h\in\Delta\backslash\{h_{1+}\}$$ is either of the form $$\forall f\in D([0,1]) \hspace{6mm} h_{c+}(f) = f(c+) = \lim_{x\to c+} f(x)$$ for some $$c\in [0,1)$$ or of the form $$\forall f\in D([0,1]) \hspace{6mm} h_{c-}(f) = f(c-) = \lim_{x\to c-} f(x)$$ for some $$c\in (0,1]$$. Let $$K=\{(c,1):c\in[0,1]\}\cup\{(c,-1):c\in(0,1]\}$$ with the weak parallel line topology . It is relatively straightforward to show that $$\Delta$$ is homeomorphic to $$K$$. $$K$$ is a separable, compact, Hausdorff space.
Since $$\Delta$$ and $$K$$ are homeomorphic, $$C(\Delta)$$ and $$C(K)$$ are isometrically isomorphic as Banach spaces. Lastly, see the implications in the first page of the paper https://doi.org/10.48550/arXiv.1112.5710 : since $$K$$ is separable, the unit ball of $$(C(K))^{\ast}$$ is weak$$^{\ast}$$ separable.