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Search: All articles in the CMB digital archive with keyword preservers

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1. CMB 2013 (vol 57 pp. 364)

Li, Lei; Wang, Ya-Shu
How Lipschitz Functions Characterize the Underlying Metric Spaces
Let $X, Y$ be metric spaces and $E, F$ be Banach spaces. Suppose that both $X,Y$ are realcompact, or both $E,F$ are realcompact. The zero set of a vector-valued function $f$ is denoted by $z(f)$. A linear bijection $T$ between local or generalized Lipschitz vector-valued function spaces is said to preserve zero-set containments or nonvanishing functions if \[z(f)\subseteq z(g)\quad\Longleftrightarrow\quad z(Tf)\subseteq z(Tg),\] or \[z(f) = \emptyset\quad \Longleftrightarrow\quad z(Tf)=\emptyset,\] respectively. Every zero-set containment preserver, and every nonvanishing function preserver when $\dim E =\dim F\lt +\infty$, is a weighted composition operator $(Tf)(y)=J_y(f(\tau(y)))$. We show that the map $\tau\colon Y\to X$ is a locally (little) Lipschitz homeomorphism.

Keywords:(generalized, locally, little) Lipschitz functions, zero-set containment preservers, biseparating maps
Categories:46E40, 54D60, 46E15

2. CMB 2010 (vol 54 pp. 141)

Kim, Sang Og; Park, Choonkil
Linear Maps on $C^*$-Algebras Preserving the Set of Operators that are Invertible in $\mathcal{A}/\mathcal{I}$
For $C^*$-algebras $\mathcal{A}$ of real rank zero, we describe linear maps $\phi$ on $\mathcal{A}$ that are surjective up to ideals $\mathcal{I}$, and $\pi(A)$ is invertible in $\mathcal{A}/\mathcal{I}$ if and only if $\pi(\phi(A))$ is invertible in $\mathcal{A}/\mathcal{I}$, where $A\in\mathcal{A}$ and $\pi:\mathcal{A}\to\mathcal{A}/\mathcal{I}$ is the quotient map. We also consider similar linear maps preserving zero products on the Calkin algebra.

Keywords:preservers, Jordan automorphisms, invertible operators, zero products
Categories:47B48, 47A10, 46H10

3. CMB 2005 (vol 48 pp. 267)

Rodman, Leiba; Šemrl, Peter; Sourour, Ahmed R.
Continuous Adjacency Preserving Maps on Real Matrices
It is proved that every adjacency preserving continuous map on the vector space of real matrices of fixed size, is either a bijective affine tranformation of the form $ A \mapsto PAQ+R$, possibly followed by the transposition if the matrices are of square size, or its range is contained in a linear subspace consisting of matrices of rank at most one translated by some matrix $R$. The result extends previously known theorems where the map was assumed to be also injective.

Keywords:adjacency of matrices, continuous preservers, affine transformations
Categories:15A03, 15A04.

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