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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...

Given a unital C*-algebra $$𝒜$$ and a right C*-module $$𝒳$$ over $$𝒜$$ , we consider the problem of finding short smooth curves in the sphere $${𝒮}_{𝒳}$$ = x ∈ $$𝒳$$ : 〈x, x〉 = 1. Curves in $${𝒮}_{𝒳}$$ are measured considering the Finsler metric which consists of the norm of $$𝒳$$ at each tangent space of $${𝒮}_{𝒳}$$ . The initial value problem is solved, for the case when $$𝒜$$ is a von Neumann algebra and $$𝒳$$ is selfdual: for any element x 0 ∈ $${𝒮}_{𝒳}$$ and any tangent vector ν at x 0, there exists a curve γ(t) = e tZ(x 0), Z ∈ $${ℒ}_{𝒜}\left(𝒳\right)$$ , Z* = −Z and ∥Z∥ ≤ π, such...

Let $U_c\left(\right)$ = u: u unitary and u-1 compact stand for the unitary Fredholm group. We prove the following convexity result. Denote by $d_{\infty }$ the rectifiable distance induced by the Finsler metric given by the operator norm in $U_c\left(\right)$. If $u₀,u₁,u\in U_c\left(\right)$ and the geodesic β joining u₀ and u₁ in $U_c\left(\right)$ satisfy $d_{\infty }(u,\beta )<\pi /2$, then the map $f\left(s\right)=d_{\infty }(u,\beta \left(s\right))$ is convex for s ∈ [0,1]. In particular, the convexity radius of the geodesic balls in $U_c\left(\right)$ is π/4. The same convexity property holds in the p-Schatten unitary groups $U_p\left(\right)$ = u: u unitary and u-1 in the p-Schatten class...

Let ${\mathcal{I}}_{\left(\mathrm{\infty }\right)}$ be the set of partial isometries with finite rank of an infinite dimensional Hilbert space $\mathcal{H}$. We show that ${\mathcal{I}}_{\left(\mathrm{\infty }\right)}$ is a smooth submanifold of the Hilbert space ${\mathcal{B}}_2\left(\mathcal{H}\right)$ of Hilbert-Schmidt operators of $\mathcal{H}$ and that each connected component is the set ${\mathcal{I}}_N$, which consists of all partial isometries of rank $N<\mathrm{\infty }$. Furthermore, ${\mathcal{I}}_{\left(\mathrm{\infty }\right)}$ is a homogeneous space of ${\mathcal{U}}_{\left(\mathrm{\infty }\right)}\times {\mathcal{U}}_{\left(\mathrm{\infty }\right)}$, where ${\mathcal{U}}_{\left(\mathrm{\infty }\right)}$ is the classical Banach-Lie group of unitary operators of $\mathcal{H}$, which are Hilbert-Schmidt perturbations of the identity. We introduce two Riemannian metrics...

Let Ω be an open subset of ℝⁿ. Let L² = L²(Ω,dx) and H¹₀ = H¹₀(Ω) be the standard Lebesgue and Sobolev spaces of complex-valued functions. The aim of this paper is to study the group of invertible operators on H¹₀ which preserve the L²-inner product. When Ω is bounded and ∂Ω is smooth, this group acts as the intertwiner of the H¹₀ solutions of the non-homogeneous Helmholtz equation u - Δu = f, ${u|}_{\partial \Omega }=0$. We show that is a real Banach-Lie group, whose Lie algebra is (i times) the space of symmetrizable operators....

For a fixed n > 2, we study the set Λ of generalized idempotents, which are operators satisfying T n+1 = T. Also the subsets Λ†, of operators such that T n−1 is the Moore-Penrose pseudo-inverse of T, and Λ*, of operators such that T n−1 = T* (known as generalized projections) are studied. The local smooth structure of these sets is examined.

Let 𝓔 be a Banach space contained in a Hilbert space 𝓛. Assume that the inclusion is continuous with dense range. Following the terminology of Gohberg and Zambickiĭ, we say that a bounded operator on 𝓔 is a proper operator if it admits an adjoint with respect to the inner product of 𝓛. A proper operator which is self-adjoint with respect to the inner product of 𝓛 is called symmetrizable. By a proper subspace 𝓢 we mean a closed subspace of 𝓔 which is the range of a proper projection. Furthermore,...