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1. CMB 2013 (vol 57 pp. 364)
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)
Linear Maps on $C^*$-Algebras Preserving the Set of Operators that are Invertible in $\mathcal{A}/\mathcal{I}$ |
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)
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. |