A nonsmooth exponential

Esteban Andruchow¹

¹ Instituto de Ciencias Universidad Nacional de Gral. Sarmiento J. M. Gutierrez entre J. L. Suarez y Verdi (1613) Los Polvorines, Argentina

Studia Mathematica, Tome 155 (2003) no. 3, pp. 265-271

Résumé

Let ${\cal M}$ be a type II$_1$ von Neumann algebra, $\tau $ a trace in ${\cal M}$, and $L^2({\cal M},\tau )$ the GNS Hilbert space of $\tau $. If $L^2({\cal M},\tau )_+$ is the completion of the set ${\cal M}_{\rm sa}$ of selfadjoint elements, then each element $\xi \in L^2({\cal M},\tau )_+$ gives rise to a selfadjoint unbounded operator $L_\xi $ on $L^2({\cal M},\tau )$. In this note we show that the exponential $\mathop {\rm exp}\nolimits :L^2({\cal M},\tau )_+ \to L^2({\cal M},\tau )$, $\mathop {\rm exp}\nolimits (\xi )=e^{iL_\xi }$, is continuous but not differentiable. The same holds for the Cayley transform $C(\xi )=(L_\xi -i)(L_\xi +i)^{-1}$. We also show that the unitary group $U_{\cal M}\subset L^2({\cal M},\tau )$ with the strong operator topology is not an embedded submanifold of $L^2({\cal M},\tau )$, in any way which makes the product $(u,w) \mapsto uw$ ($u,w\in U_{\cal M}$) a differentiable map.

Détail

DOI : 10.4064/sm155-3-5

Keywords: cal type von neumann algebra tau trace cal cal tau gns hilbert space tau cal tau completion set cal selfadjoint elements each element cal tau gives rise selfadjoint unbounded operator cal tau note exponential mathop exp nolimits cal tau cal tau mathop exp nolimits continuous differentiable holds cayley transform i unitary group cal subset cal tau strong operator topology embedded submanifold cal tau which makes product mapsto cal differentiable map