# Is there a period ring B_dif?

Let $$K$$ be a finite extension of $$\mathbb{Q}_p$$ and let $$V$$ be a p-adic representation of $$\mathrm{Gal}(\overline{\mathbb{Q}}_p/K)$$. Write $$K_\infty=K(\mu_{p^{\infty}})$$ for the cyclotomic extension of $$K$$. A construction of Sen attaches to $$V$$ a $$K_\infty$$-vector space $$\mathrm{D}_\mathrm{Sen}(V)$$ of dimension $$\mathrm{dim}_{\mathbb{Q}_p}(V)$$; it is defined as the union of all finite dimensional $$\mathrm{Gal}(K_\infty/K)$$-stable subspaces of $$(\mathbb{C}_p\otimes V)^{\mathrm{Gal}(\overline{K}/K_\infty)}$$.

Later, Fontaine similarly defined a $$K_{\infty}((t))$$-vector space $$\mathrm{D}_\mathrm{dif}(V)$$, by letting $$\mathrm{D}^+_\mathrm{dif}(V)$$ be the union of all finite rank $$\mathrm{Gal}(K_\infty/K)$$-stable $$K_\infty[[t]]$$-submodules of $$(\mathrm{B}_{\mathrm{dR}}^+\otimes V)^{\mathrm{Gal}(\overline{K}/K_\infty)}$$ and $$\mathrm{D}^+_\mathrm{dif}(V)=\mathrm{D}^+_\mathrm{dif}(V)\otimes_{K_\infty[[t]]}K_\infty((t))$$.

In his paper "Sur un resultat de S. Sen", Colmez constructs a period ring $$\mathrm{B}_{\mathrm{Sen}}$$ such that $$(\mathrm{B}_{\mathrm{Sen}}\otimes_{\mathbb{Q}_p} V)^{\mathrm{Gal}(\overline{K}/K_\infty)}$$ is isomorphic to $$\mathrm{D}_\mathrm{Sen}(V)$$. In fact $$\mathrm{B}_{\mathrm{Sen}}$$ is the ring $$\mathbb{C}_p\{\{{\log{t}\}\}}$$ of power series with nonzero radius of convergence in the variable $$\log{t}$$, where $$t$$ is Fontaine's period.

Does there exist a similar ring $$\mathrm{B}_{\mathrm{dif}}$$ which would be a period ring for $$\mathrm{D}_\mathrm{dif}$$? If so, what is it?

Thanks!