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 $L_{\xi }$ on L²(ℳ,τ). In this note we show that the exponential exp: L²(ℳ,τ)₊ → L²(ℳ,τ), $exp\left(\xi \right)=e^{i{L}_{\xi }}$, is continuous but not differentiable. The same holds for the Cayley transform $C\left(\xi \right)=(L_{\xi }-i){(L_{\xi }+i)}^{-1}$. We also show that the unitary group $U_{ℳ}\subset L²(ℳ,\tau )$ with the strong operator topology is not an embedded submanifold...