A nonsmooth exponential

• Autorzy: Esteban Andruchow
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 58B25: Group structures and generalizations on infinite-dimensional manifolds
• 46L51: Noncommutative measure and integration
• 58B10: Differentiability questions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2003
• Tom: 155
• Numer: 3
• Strony: 265-271

Daty

wydano

2003

Twórcy

autor

Esteban Andruchow

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

Identyfikatory

DOI

10.4064/sm155-3-5