arXiv subjects

Esteban Andruchow

Publications and source records attributed to Esteban Andruchow.

40 records · Page 3Linked to original sources

arXiv2008

Lagrangian Grassmannian in Infinite Dimension

Given a complex structure $J$ on a real (finite or infinite dimensional) Hilbert space $H$, we study the geometry of the Lagrangian Grassmannian $Λ(H)$ of $H$, i.e. the set of closed linear subspaces $L\subset H$ such that $$J(L)=L^\perp.$$ The complex unitary group $U(H_J)$, consisting of the elements of the orthogonal group of $H$ which are complex linear for the given complex structure, acts transitively on $Λ(H)$ and induces a natural linear connection in $Λ(H)$. It is shown that any pair of Lagrangian subspaces can be joined by a geodesic of this connection. A Finsler metric can also be introduced, if one regards subspaces $L$ as projections $p_L$ (=the orthogonal projection onto $L$) or symmetries $\e_L=2p_L-I$, namely measuring tangent vectors with the operator norm. We show that for this metric the Hopf-Rinow theorem is valid in $Λ(H)$: a geodesic joining a pair of Lagrangian subspaces can be chosen to be of minimal length. We extend these results to the classical Banach-Lie groups of Schatten.

math.DG↗

arXiv2008

Nonpositively curved metric in the positive cone of a finite von Neumann algebra

In this paper we study the metric geometry of the space $Σ$ of positive invertible elements of a von Neumann algebra ${\mathcal A}$ with a finite, normal and faithful tracial state $τ$. The trace induces an incomplete Riemannian metric $ _a=τ(ya^{-1}xa^{-1})$, and though the techniques involved are quite different, the situation here resembles in many relevant aspects that of the $n\times n$ matrices when they are regarded as a symmetric space. For instance we prove that geodesics are the shortest paths for the metric induced, and that the geodesic distance is a convex function; we give an intrinsic (algebraic) characterization of the geodesically convex submanifolds $M$ of $Σ$, and under suitable hypothesis we prove a factorization theorem for elements in the algebra that resembles the Iwasawa decomposition for matrices. This factorization is obtained \textit{via} a nonlinear orthogonal projection $Π_M:Σ\to M$, a map which turns out to be contractive for the geodesic distance.

math.DG↗

arXiv2008

Connes' metric for states in group algebras

In this article we follow the main idea of A. Connes for the construction of a metric in the state space of a C*-algebra. We focus in the reduced algebra of a discrete group $Γ$, and prove some equivalences and relations between two central objects of this category: the word-length growth (connected with the degree of the extension of $Γ$ when the group is an extension of Z by a finite group), and the topological equivalence between the w*-topology and the one introduced with this metric in the state space of $C_r*(Γ)$.

math.OA↗

arXiv2000

Homotopy of vector states

Let $B$ be a C$^*$-algebra and $X$ a C$^*$ Hilbert $B$-module. If $p\in B$ is a projection, denote by $S_p =\{x\in X : < x,x> =p\}$, the $p$-sphere of $X$. For $ϕ$ a state of $B$ with support $p$ in $B$ and $x\in S_p$, consider the state $ϕ_x$ of $L_B(X)$ given by $ϕ_x(t)= ϕ(< x,t(x)>)$. In this paper we study certain sets associated to these states, and examine their topologic properties. As an application of these techniques, we prove that the space of states of the hyperfinite II$_1$ factor $R_0$, with support equivalent to a given projection $p\in R_0$, regarded with the norm topology (of the conjugate space of $R_0$), has trivial homotopy groups of all orders. The same holds for the space $$ S_p(R_0)=\{v\in R_0:v^*v=p\}\subset R_0 $$ of partial isometries with initial space $p$, regarded with the ultraweak topology.

math.OA↗