scholarly journals Generalized Grassmann graphs associated to conjugacy classes of finite-rank self-adjoint operators

2021 ◽  
Author(s):  
Mark Pankov ◽  
Krzysztof Petelczyc ◽  
Mariusz Żynel
Keyword(s):  
Finite Rank ◽  
Conjugacy Classes ◽  
Adjoint Operators

scholarly journals Automorphisms and Some Geodesic Properties of Ortho-Grassmann Graphs

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Mark Pankov ◽  
Krzysztof Petelczyc ◽  
Mariusz Źynel
Keyword(s):  
Hilbert Space ◽  
Finite Rank ◽  
Unitary Operator ◽  
Complex Hilbert Space ◽  
Conjugacy Classes ◽  
Grassmann Graph ◽  
Rank 2 ◽  
Adjoint Operators

Let $H$ be a complex Hilbert space. Consider the ortho-Grassmann graph $\Gamma^{\perp}_{k}(H)$ whose vertices are $k$-dimensional subspaces of $H$ (projections of rank $k$) and two subspaces are connected by an edge in this graph if they are compatible and adjacent (the corresponding rank-$k$ projections commute and their difference is an operator of rank $2$). Our main result is the following: if $\dim H\ne 2k$, then every automorphism of $\Gamma^{\perp}_{k}(H)$ is induced by a unitary or anti-unitary operator; if $\dim H=2k\ge 6$, then every automorphism of $\Gamma^{\perp}_{k}(H)$ is induced by a unitary or anti-unitary operator or it is the composition of such an automorphism and the orthocomplementary map. For the case when $\dim H=2k=4$ the statement fails. To prove this statement we compare geodesics of length two in ortho-Grassmann graphs and characterise compatibility (commutativity) in terms of geodesics in Grassmann and ortho-Grassmann graphs. At the end, we extend this result on generalised ortho-Grassmann graphs associated to conjugacy classes of finite-rank self-adjoint operators.

An infinite superstable group has infinitely many conjugacy classes

1991 ◽  
Vol 56 (2) ◽  
pp. 618-623 ◽  
Author(s):  
I. Aguzarov ◽  
R. E. Farey ◽  
J. B. Goode
Keyword(s):  
Simple Group ◽  
Finite Rank ◽  
Conjugacy Classes ◽  
Morley Rank ◽  
Stable Group ◽  
The Third ◽  
Rank One ◽  
Preliminary Version ◽  
Finite Morley Rank ◽  
Basic Reference

We begin with some notes concerning the genesis of this paper. A preliminary version of it was written by the third author, who was moved by the desire to correct a mistake in Poizat [1987, p. 97], and to refresh some other minor results of the same book, concerning equations which are satisfied generically in a stable group. The book in question will be considered here as our basic reference on stable groups, and these other results will be discussed elsewhere.This preliminary version contained §§1, 2 and 3 of the present paper, restricted to the context of groups of finite Morley rank. It was observed that a counterexample of finite rank to the above theorem would be an extreme refutation of a conjecture by Zil'ber and Cherlin (cyrillic alphabet order), that may be not so solid as was believed some time ago, which states that a simple group of finite rank should be an algebraic group. Additional motivation for the problem was seen in Reineke's theorem (Reineke [1975]), stating that a connected group of rank one is abelian—the cornerstone for the study of superstable groups—whose proof rests on the fact that a group with two conjugacy classes either has only two elements, or has infinite chains of centralizers (a property that violates stability).

scholarly journals Matrix measures and finite rank perturbations of self-adjoint operators

2020 ◽  
Vol 10 (4) ◽  
pp. 1173-1210 ◽  
Author(s):  
Constanze Liaw ◽  
Sergei Treil
Keyword(s):  
Finite Rank ◽  
Adjoint Operators ◽  
Matrix Measures

scholarly journals Dimension of the Exceptional Set in the Aronszajn–Donoghue Theorem for Finite Rank Perturbations

2020 ◽  
Author(s):  
Constanze Liaw ◽  
Sergei Treil ◽  
Alexander Volberg
Keyword(s):  
Finite Rank ◽  
Measure Zero ◽  
Adjoint Operator ◽  
Cyclic Vector ◽  
Spectral Measures ◽  
Hermitian Matrices ◽  
Exceptional Set ◽  
Direct Sum ◽  
Rank One Perturbation ◽  
Rank One

Abstract The classical Aronszajn–Donoghue theorem states that for a rank-one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $d$ perturbations $A_{\boldsymbol{\alpha }}:= A + {\textbf{B}} {\boldsymbol{\alpha }} {\textbf{B}}^*$, ${\textbf{B}}:{\mathbb C}^d\to{{\mathcal{H}}}$, with ${\operatorname{Ran}}{\textbf{B}}$ being cyclic for $A$, parametrized by $d\times d$ Hermitian matrices ${\boldsymbol{\alpha }}$, the singular parts of the spectral measures of $A$ and $A_{\boldsymbol{\alpha }}$ are mutually singular for all ${\boldsymbol{\alpha }}$ except for a small exceptional set $E$. It was shown earlier by the 1st two authors, see [4], that $E$ is a subset of measure zero of the space $\textbf{H}(d)$ of $d\times d$ Hermitian matrices. In this paper, we show that the set $E$ has small Hausdorff dimension, $\dim E \le \dim \textbf{H}(d)-1 = d^2-1$.

scholarly journals Coarse median algebras: the intrinsic geometry of coarse median spaces and their intervals

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Graham A. Niblo ◽  
Nick Wright ◽  
Jiawen Zhang
Keyword(s):  
Finite Rank ◽  
Intrinsic Geometry ◽  
Bounded Geometry ◽  
Higher Dimensional ◽  
Median Algebra ◽  
Asymptotic Geometry ◽  
Median Operator

AbstractThis paper establishes a new combinatorial framework for the study of coarse median spaces, bridging the worlds of asymptotic geometry, algebra and combinatorics. We introduce a simple and entirely algebraic notion of coarse median algebra which simultaneously generalises the concepts of bounded geometry coarse median spaces and classical discrete median algebras. We study the coarse median universe from the perspective of intervals, with a particular focus on cardinality as a proxy for distance. In particular we prove that the metric on a quasi-geodesic coarse median space of bounded geometry can be constructed up to quasi-isometry using only the coarse median operator. Finally we develop a concept of rank for coarse median algebras in terms of the geometry of intervals and show that the notion of finite rank coarse median algebra provides a natural higher dimensional analogue of Gromov’s concept of $$\delta $$ δ -hyperbolicity.

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