arXiv subjects

Esteban Andruchow

Publications and source records attributed to Esteban Andruchow.

At least 37 records · Page 2Linked to original sources

arXiv2018

Projective geometry in the Poincaré disk of a $C^*$-algebra

We study the Poincaré disk ${\cal D}=\{a\in {\cal A}: \|a\|<1\}$ of a C$^*$-algebra ${\cal A}$ from a projective point of view: ${\cal D}$ is regarded as an open subset of the projective line $\mathbb{P}_1{\cal A}$, the space of complemented rank one submodules of ${\cal A}^2$. We introduce the concept of cross ratio of four points in $\mathbb{P}_1{\cal A}$. Our main result establishes the relation between the exponential map $Exp_{z_0}(z_1)$ of ${\cal D}$ ($z_0,z_1\in {\cal D}$) and the cross ratio of the four-tuple $$ δ(-\infty), δ(0)=z_0, δ(1)=z_1 , δ(+\infty), $$ where $δ$ is the unique geodesic of ${\cal D}$ joining $z_0$ and $z_1$ at times $t=0$ and $t=1$, respectively.

math.OA↗

arXiv2018

Projections with fixed difference: a Hopf-Rinow theorem

The set $D_{A_0}$, of pairs of orthogonal projections $(P,Q)$ in generic position with fixed difference $P-Q=A_0$, is shown to be a homogeneus smooth manifold: it is the quotient of the unitary group of the commutant $\{A_0\}'$ divided by the unitary subgroup of the commutant $\{P_0, Q_0\}'$, where $(P_0,Q_0)$ is any fixed pair in $D_{A_0}$. Endowed with a natural reductive structure (a linear connection) and the quotient Finsler metric of the operator norm, it behaves as a classic Riemannian space: any two pairs in $D_{A_0}$ are joined by a geodesic of minimal length. Given a base pair $(P_0,Q_0)$, pairs in an open dense subset of $D_{A_0}$ can be joined to $(P_0,Q_0)$ by a {\it unique} minimal geodesic.

math.FA↗

arXiv2018

Canonical sphere bundles of the Grassmann manifold

For a given Hilbert space $\mathcal H$, consider the space of self-adjoint projections $\mathcal P(\mathcal H)$. In this paper we study the differentiable structure of a canonical sphere bundle over $\mathcal P(\mathcal H)$ given by $$ \mathcal R=\{\, (P,f)\in \mathcal P(\mathcal H)\times \mathcal H \, : \, Pf=f , \, \|f\|=1\, \}. $$ We establish the smooth action on $\mathcal R$ of the group of unitary operators of $\mathcal H$, therefore $\mathcal R$ is an homogeneous space. Then we study the metric structure of $\mathcal R$ by endowing it first with the uniform quotient metric, which is a Finsler metric, and we establish minimality results for the geodesics. These are given by certain one-parameter groups of unitary operators, pushed into $\mathcal R$ by the natural action of the unitary group. Then we study the restricted bundle $\mathcal R_2^+$ given by considering only the projections in the restricted Grassmannian, locally modelled by Hilbert-Schmidt operators. Therefore we endow $\mathcal R_2^+$ with a natural Riemannian metric that can be obtained by declaring that the action of the group is a Riemannian submersion. We study the Levi-Civita connection of this metric and establish a Hopf-Rinow theorem for $\mathcal R_2^+$, again obtaining a characterization of the geodesics as the image of certain one-parameter groups with special speeds.

math.DG↗

arXiv2017

Poncaré half-space of a C*-algebra

Let $A$ be a C$*^$-algebra. Given a representation $A\subset B(L)$ in a Hilbert space $L$, the set $G^+\subset A$ of positive invertible elements can be thought as the set of inner products in $L$, related to $A$, which are equivalent to the original inner product. The set $G^+$ has a rich geometry, it is a homogeneous space of the invertible group $G$ of $A$, with an invariant Finsler metric. In the present paper we study the tangent bundle $TG^+$ of $G^+$, as a homogenous Finsler space of a natural group of invertible matrices in $M_2(A)$, identifying $TG^+$ with the {\it Poincaré halfspace} $H$ of $A$, $$ H=\{h\in A: Im(h)\ge 0, Im(h) \hbox{ invertible}\}. $$ We show that $\h\simeq TG^+$ has properties similar to those of a space of non-positive constant curvature.

math.OA↗

arXiv2017

Schmidt decomposable products of projections

We characterize operators $T=PQ$ ($P,Q$ orthogonal projections in a Hilbert space $H$) which have a singular value decomposition. A spatial characterizations is given: this condition occurs if and only if there exist orthonormal bases $\{ψ_n\}$ of $R(P)$ and $\{ξ_n\}$ of $R(Q)$ such that $\langleξ_n,ψ_m\rangle=0$ if $n\ne m$. Also it is shown that this is equivalent to $A=P-Q$ being diagonalizable. Several examples are studied, relating Toeplitz, Hankel and Wiener-Hopf operators to this condition. We also examine the relationship with the differential geometry of the Grassmann manifold of underlying the Hilbert space: if $T=PQ$ has a singular value decomposition, then the generic parts of $P$ and $Q$ are joined by a minimal geodesic with diagonalizable exponent.

math.FA↗

arXiv2017

Uncertainty principle and geometry of the infinite Grassmann manifold

We study the pairs of projections $$ P_If=χ_If ,\ \ Q_Jf= \left(χ_J \hat{f}\right)\check{\ } , \ \ f\in L^2(\mathbb{R}^n), $$ where $I, J\subset \mathbb{R}^n$ are sets of finite Lebesgue measure, $χ_I, χ_J$ denote the corresponding characteristic functions and $\hat{\ } , \check{\ }$ denote the Fourier-Plancherel transformation $L^2(\mathbb{R}^n)\to L^2(\mathbb{R}^n)$ and its inverse. These pairs of projections have been widely studied by several authors in connection with the mathematical formulation of Heisenberg's uncertainty principle. Our study is done from a differential geometric point of view. We apply known results on the Finsler geometry of the Grassmann manifold ${\cal P}({\cal H})$ of a Hilbert space ${\cal H}$ to establish that there exists a unique minimal geodesic of ${\cal P}({\cal H})$, which is a curve of the form $$ δ(t)=e^{itX_{I,J}}P_Ie^{-itX_{I,J}} $$ which joins $P_I$ and $Q_J$ and has length $π/2$. As a consequence we obtain that if $H$ is the logarithm of the Fourier-Plancherel map, then $$ \|[H,P_I]\|\ge π/2. $$ The spectrum of $X_{I,J}$ is denumerable and symmetric with respect to the origin, it has a smallest positive eigenvalue $γ(X_{I,J})$ which satisfies $$ \cos(γ(X_{I,J}))=\|P_IQ_J\|. $$

math.FA↗

arXiv2017

Essentially orthogonal subspaces

We study the set ${\cal C}$ consisting of pairs of orthogonal projections $P,Q$ acting in a Hilbert space ${\cal H}$ such that $PQ$ is a compact operator. These pairs have a rich geometric structure which we describe here. They are parted in three subclasses: ${\cal C}_0$ which consists of pairs where $P$ or $Q$ have finite rank, ${\cal C}_1$ of pairs such that $Q$ lies in the restricted Grassmannian (also called Sato Grassmannian) of the polarization ${\cal H}=N(P)\oplus R(P)$, and ${\cal C}_\infty$. Belonging to this last subclass one has the pairs $$ P_If=χ_If ,\ \ Q_Jf= \left(χ_J \hat{f}\right)\check{\ } , \ \ f\in L^2(\mathbb{R}^n), $$ where $I, J\subset \mathbb{R}^n$ are sets of finite Lebesgue measure, $χ_I, χ_J$ denote the corresponding characteristic functions and $\hat{\ } , \check{\ }$ denote the Fourier-Plancherel transform $L^2(\mathbb{R}^2)\to L^2(\mathbb{R}^2)$ and its inverse. We characterize the connected components of these classes: the components of ${\cal C}_0$ are parametrized by the rank, the components of ${\cal C}_1$ are parametrized by the Fredholm index of the pairs, and ${\cal C}_\infty$ is connected. We show that these subsets are (non complemented) differentiable submanifolds of ${\cal B}({\cal H})\times {\cal B}({\cal H})$.

math.FA↗

arXiv2016

Geometric significance of Toeplitz kernels

Let $L^2$ be the Lebesgue space of square-integrable functions on the unit circle. We show that the injectivity problem for Toeplitz operators is linked to the existence of geodesics in the Grassmann manifold of $L^2$. We also investigate this connection in the context of restricted Grassmann manifolds associated to $p$-Schatten ideals and essentially commuting projections.

math.FA↗

arXiv2015

Proper subspaces and compatibility

Let $\mathcal{E}$ be a Banach space contained in a Hilbert space $\mathcal{L}$. Assume that the inclusion is continuous with dense range. Following the terminology of Gohberg and Zambicki\vı, we say that a bounded operator on $\mathcal{E}$ is a proper operator if it admits an adjoint with respect to the inner product of $\mathcal{L}$. By a proper subspace $\mathcal{S}$ we mean a closed subspace of $\mathcal{E}$ which is the range of a proper projection. If there exists a proper projection which is also self-adjoint with respect to the inner product of $\mathcal{L}$, then $\mathcal{S}$ belongs to a well-known class of subspaces called compatible subspaces. We find equivalent conditions to describe proper subspaces. Then we prove a necessary and sufficient condition to ensure that a proper subspace is compatible. Each proper subspace $\mathcal{S}$ has a supplement $\mathcal{T}$ which is also a proper subspace. We give a characterization of the compatibility of both subspaces $\mathcal{S}$ and $\mathcal{T}$. Several examples are provided that illustrate different situations between proper and compatible subspaces.

math.FA↗

arXiv2012

The group of L^2 isometries on H^1_0

Let U be an open subset of R^n. Let L^2=L^2(U,dx) and H^1_0=H^1_0(U) be the standard Lebesgue and Sobolev spaces of complex-valued functions. The aim of this paper is to study the group G of invertible operators on H^1_0 which preserve the L^2-inner product. When U is bounded and the border $\partial U$ is smooth, this group acts as the intertwiner of the H^1_0 solutions of the non-homogeneous Helmholtz equation $u-Δu=f$, $u|_{\partial U}=0$. We show that G is a real Banach-Lie group, whose Lie algebra is (i times) the space of symmetrizable operators. We discuss the spectrum of operators belonging to G by means of examples. In particular, we give an example of an operator in G whose spectrum is not contained in the unit circle. We also study the one parameter subgroups of G. Curves of minimal length in G are considered. We introduce the subgroups G_p:=G \cap (I - B_p(H^1_0)), where B_p(H_0^1) is a Schatten ideal of operators on H_0^1. An invariant (weak) Finsler metric is defined by the p-norm of the Schatten ideal of operators of L^2. We prove that any pair of operators g_1,g_2 in G_p can be joined by a minimal curve of the form $a(t)=g_1 e^{itX}$, where X is a symmetrizable operator in B_p(H^1_0).

math.FA↗

arXiv2011

The left invariant metric in the general linear group

Left invariant metrics induced by the p-norms of the trace in the matrix algebra are studied on the general lineal group. By means of the Euler-Lagrange equations, existence and uniqueness of extremal paths for the length functional are established, and regularity properties of these extremal paths are obtained. Minimizing paths in the group are shown to have a velocity with constant singular values and multiplicity. In several special cases, these geodesic paths are computed explicitly. In particular the Riemannian geodesics, corresponding to the case p=2, are characterized as the product of two one-parameter groups. It is also shown that geodesics are one-parameter groups if and only if the initial velocity is a normal matrix. These results are further extended to the context of compact operators with p-summable spectrum, where a differential equation for the spectral projections of the velocity vector of an extremal path is obtained.

math.DG↗

arXiv2011

A characterization of Hermitian matrices with variable diagonal and smallest operator norm

We describe properties of a Hermitian square matrix M in M_n(C) equivalent to that of having minimal quotient norm in the following sense: ||M|| <= ||M+D|| for all real diagonal matrices D in M_n(C) and || || the operator norm. These matrices are related to some particular positive matrices with their range included in the eigenspaces of the eigenvalues +||M|| and -||M|| of M. We show how a constructive method can be used to obtain minimal matrices of any dimension relating this problem with majorization results in R^n.

math.OA↗

arXiv2010

Smooth paths of conditional expectations

Let A be a von Neumann algebra with a finite trace $τ$, represented in $H=L^2(A,τ)$, and let $B_t\subset A$ be sub-algebras, for $t$ in an interval $I$. Let $E_t:A\to B_t$ be the unique $τ$-preserving conditional expectation. We say that the path $t\mapsto E_t$ is smooth if for every $a\in A$ and $v \in H$, the map $$ I\ni t\mapsto E_t(a)v\in H $$ is continuously differentiable. This condition implies the existence of the derivative operator $$ dE_t(a):H\to H, \ dE_t(a)v=\frac{d}{dt}E_t(a)v. $$ If this operator verifies the additional boundedness condition, $$ \int_J \|dE_t(a)\|_2^2 d t\le C_J\|a\|_2^2, $$ for any closed bounded sub-interval $J\subset I$, and $C_J>0$ a constant depending only on $J$, then the algebras $B_t$ are *-isomorphic. More precisely, there exists a curve $G_t:A\to A$, $t\in I$ of unital, *-preserving linear isomorphisms which intertwine the expectations, $$ G_t\circ E_0=E_t\circ G_t. $$ The curve $G_t$ is weakly continuously differentiable. Moreover, the intertwining property in particular implies that $G_t$ maps $B_0$ onto $B_t$. We show that this restriction is a multiplicative isomorphism.

math.OA↗

arXiv2009

Homogeneous manifolds from noncommutative measure spaces

Let M be a finite von Neumann algebra with a faithful trace $τ$. In this paper we study metric geometry of homogeneous spaces O of the unitary group U of M, endowed with a Finsler quotient metric induced by the p-norms of $τ$, $||x||_p=τ(|x|^p)^{1/p}$, $p\ge 1$. The main results include the following. The unitary group carries on a rectifiable distance d_p induced by measuring the length of curves with the p-norm. If we identify O as a quotient of groups, then there is a natural quotient distance d'_p that metrizes the quotient topology. On the other hand, the Finsler quotient metric defined in O provides a way to measure curves, and therefore, there is an associated rectifiable distance d_{O,p}. For $p\ge 2$, we prove that the distances d'_p and d_{O,p} coincide. Based on this fact, we show that the metric space (O,d'_p) is a complete path metric space. The other problem treated in this article is the existence of metric geodesics, or curves of minimal length, in O. We give two abstract partial results in this direction. The first concerns the initial values problem and the second the fixed endpoints problem. We show how these results apply to several examples. In the process, we improve some results about the metric geometry of U with the p-norm.

math.MG↗

arXiv2008

The rectifiable distance in the unitary Fredholm group

Let $U_c(H)={u: u is unitary and u-1 is compact}$ 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 $U_c(H)$. If $u_0,u_1,u\in U_c(H)$ and the geodesic $β$ joining $u_0$ and $u_1$ in $U_c(H)$ verifies $d_\infty(u,β)<π/2$, then the map $f(s)=d_\infty(u,β(s))$ is convex for $s\in[0,1]$. In particular the convexity radius of the geodesic balls in $U_c(H)$ is $π/4$. The same convexity property holds in the $p$-Schatten unitary groups $U_p(H)={u: u is unitary and u-1 is in the p-Schatten class}$, for $p$ an even integer, $p\ge 4$ (in this case, the distance is strictly convex). The same results hold in the unitary group of a $C^*$-algebra with a faithful finite trace. We apply this convexity result to establish the existence of curves of minimal length with given initial conditions, in the unitary orbit of an operator, under the action of the Fredholm group. We characterize self-adjoint operators $A$ such that this orbit is a submanifold (of the affine space $A+K(H)$, where $K(H)$=compact operators).

math.OA↗

arXiv2008

Hopf-Rinow Theorem in the Sato Grassmannian

Let $U_2({\cal H})$ be the Banach-Lie group of unitary operators in the Hilbert space ${\cal H}$ which are Hilbert-Schmidt perturbations of the identity 1. In this paper we study the geometry of the unitary orbit $$\{upu^*: u\in U_2({\cal H})\},$$ of an infinite projection $p$ in ${\cal H}$. This orbit coincides with the connected component of $p$ in the Hilbert-Schmidt restricted Grassmannian $Gr_{res}(p)$ (also known in the literature as the Sato Grassmannian) corresponding to the polarization ${\cal H}=p({\cal H})\oplus p({\cal H})^\perp$. It is known that the components of $Gr_{res}(p)$ are differentiable manifolds. Here we give a simple proof of the fact that $Gr_{res}^0(p)$ is a smooth submanifold of the affine Hilbert space $p+{\cal B}_2({\cal H})$, where ${\cal B}_2({\cal H})$ denotes the space of Hilbert-Schmidt operators of ${\cal H}$. We prove that the geodesics of the natural connection, which are of the form $γ(t)=e^{tz}pe^{-tz}$, for $z$ a $p$-codiagonal anti-hermitic element of ${\cal B}_2({\cal H})$, have minimal length provided that $\|z\|\le π/2$. Note that the condition is given in terms of the usual operator norm, a fact which implies that there exist minimal geodesics of arbitrary length. Also we show that any two points $p_1,p_2\in Gr_{res}^0(p)$ are joined by a minimal geodesic. If moreover $\|p_1-p_2\|<1$, the minimal geodesic is unique. Finally, we replace the 2-norm by the $k$-Schatten norm ($k>2$), and prove that the geodesics are also minimal for these norms, up to a critical value of $t$, which is estimated also in terms of the usual operator norm. In the process, minimality results in the $k$-norms are also obtained for the group $U_2({\cal H})$.

math.DG↗

arXiv2008

Weak Riemannian manifolds from finite index subfactors

Let $N\subset M$ be a finite Jones' index inclusion of II$_1$ factors, and denote by $U_N\subset U_M$ their unitary groups. In this paper we study the homogeneous space $U_M/U_N$, which is a (infinite dimensional) differentiable manifold, diffeomorphic to the orbit $$ {\cal O}(p) =\{u p u^*: u\in U_M\} $$ of the Jones projection $p$ of the inclusion. We endow ${\cal O}(p) $ with a Riemannian metric, by means of the trace on each tangent space. These are pre-Hilbert spaces (the tangent spaces are not complete), therefore ${\cal O}(p)$ is a weak Riemannian manifold. We show that ${\cal O}(p)$ enjoys certain properties similar to classic Hilbert-Riemann manifolds. Among them, metric completeness of the geodesic distance, uniqueness of geodesics of the Levi-Civita connection as minimal curves, and partial results on the existence of minimal geodesics. For instance, around each point $p_1$ of ${\cal O}(p)$, there is a ball $\{q\in {\cal O}(p):\|q-p_1\|<r\}$ (of uniform radius $r$) of the usual norm of $M$, such that any point $p_2$ in the ball is joined to $p_1$ by a unique geodesic, which is shorter than any other piecewise smooth curve lying inside this ball. We also give an intrinsic (algebraic) characterization of the directions of degeneracy of the submanifold inclusion ${\cal O}(p)\subset {\cal P}(M_1)$, where the last set denotes the Grassmann manifold of the von Neumann algebra generated by $M$ and $p$.

math.DG↗

arXiv2008

Finsler geometry and actions of the p-Schatten unitary groups

Let $p$ be an even positive integer and $U_p(H)$ be the Banach-Lie group of unitary operators $u$ which verify that $u-1$ belongs to the $p$-Schatten ideal $B_p(H)$. Let ${\cal O}$ be a smooth manifold on which $U_p(H)$ acts transitively and smoothly. Then one can endow ${\cal O}$ with a natural Finsler metric in terms of the $p$-Schatten norm and the action of $U_p(H)$. Our main result establishes that for any pair of given initial conditions $$ x\in {\cal O}\hbox{and} X\in (T{\cal O})_x $$ there exists a curve $δ(t)=e^{tz}\cdot x$ in ${\cal O}$, with $z$ a skew-hermitian element in the $p$-Schatten class such that $$ δ(0)=x \hbox{and} \dotδ(0)=X, $$ which remains minimal as long as $t\|z\|_p\le π/4$. Moreover, $δ$ is unique with these properties. We also show that the metric space $({\cal O},d)$ ($d=$ rectifiable distance) is complete. In the process we establish minimality results in the groups $U_p(H)$, and a convexity property for the rectifiable distance. As an example of these spaces, we treat the case of the unitary orbit $$ {\cal O}=\{uAu^*: u\in U_p(H)\} $$ of a self-adjoint operator $A\in B(H)$.

math.DG↗