A nonsmooth exponential

Esteban Andruchow. "A nonsmooth exponential." Studia Mathematica 155.3 (2003): 265-271. <http://eudml.org/doc/284834>.

@article{EstebanAndruchow2003,
abstract = {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.},
author = {Esteban Andruchow},
journal = {Studia Mathematica},
keywords = {unitary group; noncommutative integration},
language = {eng},
number = {3},
pages = {265-271},
title = {A nonsmooth exponential},
url = {http://eudml.org/doc/284834},
volume = {155},
year = {2003},
}

TY - JOUR
AU - Esteban Andruchow
TI - A nonsmooth exponential
JO - Studia Mathematica
PY - 2003
VL - 155
IS - 3
SP - 265
EP - 271
AB - 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.
LA - eng
KW - unitary group; noncommutative integration
UR - http://eudml.org/doc/284834
ER -