For Dewitt super groups $G$ modeled via an underlying finitely generated Grassmann algebra it is well known that when there exists a body group $BG$ compatible with the group operation on $G$, then, generically, the kernel $K$ of the body homomorphism is nilpotent. This is not true when the underlying Grassmann algebra is infinitely generated. We show that it is quasi-nilpotent in the sense that as a Banach Lie group its Lie algebra $\kappa$ has the property that for each $a\,\in \,\kappa ,\,\text{a}{{\text{d}}_{a}}$ has a zero spectrum. We also show that the exponential mapping from $\kappa$ to $K$ is surjective and that $K$ is a quotient manifold of the Banach space $\kappa$ via a lattice in $\kappa$.

Let $\mathcal{M}$ be a type $\text{I}{{\text{I}}_{1}}$ von Neumann algebra, $\tau$ a trace in $\mathcal{M}$, and ${{L}^{2}}\left( \mathcal{M},\tau \right)$ the GNS Hilbert space of $\tau$. We regard the unitary group ${{U}_{\mathcal{M}}}$ as a subset of ${{L}^{2}}\left( \mathcal{M},\tau \right)$ and characterize the shortest smooth curves joining two fixed unitaries in the ${{L}^{2}}$ metric. As a consequence of this we obtain that ${{U}_{\mathcal{M}}}$, though a complete (metric) topological group, is not an embedded riemannian submanifold of ${{L}^{2}}\left( \mathcal{M},\tau \right)$

In this paper it is shown that inclusions inside the Segal-Wilson Grassmannian give rise to Darboux transformations between the solutions of the $\text{KP}$ hierarchy corresponding to these planes. We present a closed form of the operators that procure the transformation and express them in the related geometric data. Further the associated transformation on the level of $\tau $-functions is given.