A nonsmooth exponential

Esteban Andruchow

Studia Mathematica, Tome 157 (2003), p. 265-271 / Harvested from The Polish Digital Mathematics Library

Résumé

Let ℳ be a type II₁ von Neumann algebra, τ a trace in ℳ, and L²(ℳ,τ) the GNS Hilbert space of τ. If L²(ℳ,τ)₊ is the completion of the set ${ℳ}_{sa}$ of selfadjoint elements, then each element ξ ∈ L²(ℳ,τ)₊ gives rise to a selfadjoint unbounded operator $L_{ξ}$ on L²(ℳ,τ). In this note we show that the exponential exp: L²(ℳ,τ)₊ → L²(ℳ,τ), $exp(ξ) = e^{iL_{ξ}}$, is continuous but not differentiable. The same holds for the Cayley transform $C(ξ) = (L_{ξ} - i)(L_{ξ} + i)^{-1}$. We also show that the unitary group $U_{ℳ} ⊂ L²(ℳ,τ)$ with the strong operator topology is not an embedded submanifold of L²(ℳ,τ), in any way which makes the product (u,w) ↦ uw ($u,w ∈ U_{ℳ}$) a differentiable map.

Détail

Publié le : 2003-01-01

Zbl 1028.46092

EUDML-ID : urn:eudml:doc:284834