arXiv subjects

Yunbai Dong

Publications and source records attributed to Yunbai Dong.

3 recordsLinked to original sources

arXiv2021

Transformations preserving the norm of means between positive cones of general and commutative $C^*$-algebras

In this paper, we consider a (nonlinear) transformation $Φ$ of invertible positive elements in $C^*$-algebras which preserves the norm of any of the three fundamental means of positive elements; namely, $\|Φ(A)\mm Φ(B)\| = \|A\mm B\|$, where $\mm$ stands for the arithmetic mean $A\nabla B=(A+B)/2$, the geometric mean $A\#B=A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2}$, or the harmonic mean $A!B=2(A^{-1} + B^{-1})^{-1}$. Assuming that $Φ$ is surjective and preserves either the norm of the arithmetic mean or the norm of the geometric mean, we show that $Φ$ extends to a Jordan $*$-isomorphism between the underlying full algebras. If $Φ$ is surjective and preserves the norm of the harmonic mean, then we obtain the same conclusion in the special cases where the underlying algebras are $AW^*$-algebras or commutative $C^*$-algebras. In the commutative case, for a transformation $T: F(\mathrm{X})\subset C_0(\mathrm{X})_+\rightarrow C_0(\mathrm{Y})_+$, we can relax the surjectivity assumption and show that $T$ is a generalized composition operator if $T$ preserves the norm of the (arithmetic, geometric, harmonic, or in general any power) mean of any finite collection of positive functions, provided that the domain $F(\mathrm{X})$ contains sufficiently many elements to peak on compact $G_δ$ sets. When the image $T(F(\mathrm{X}))$ also contains sufficiently many elements to peak on compact $G_δ$ sets, $T$ extends to an algebra $*$-isomorphism between the underlying full function algebras.

math.OA↗

arXiv2014

Stability of Banach spaces via nonlinear $\varepsilon$-isometries

In this paper, we prove that the existence of an $\varepsilon$-isometry from a separable Banach space $X$ into $Y$ (the James space or a reflexive space) implies the existence of a linear isometry from $X$ into $Y$. Then we present a set valued mapping version lemma on non-surjective $\varepsilon$-isometries of Banach spaces. Using the above results, we also discuss the rotundity and smoothness of Banach spaces under the perturbation by $\varepsilon$-isometries.

math.FA↗

arXiv2013

Universal stability of Banach spaces for $\varepsilon$-isometries

Let $X$, $Y$ be two real Banach spaces and $\varepsilon>0$. A standard $\varepsilon$-isometry $f:X\rightarrow Y$ is said to be $(α,γ)$-stable (with respect to $T:L(f)\equiv\overline{\rm span}f(X)\rightarrow X$ for some $α, γ>0$) if $T$ is a linear operator with $\|T\|\leqα$ so that $Tf-Id$ is uniformly bounded by $γ\varepsilon$ on $X$. The pair $(X,Y)$ is said to be stable if every standard $\varepsilon$-isometry $f:X\rightarrow Y$ is $(α,γ)$-stable for some $α,γ>0$. $X (Y)$ is said to be universally left (right)-stable, if $(X,Y)$ is always stable for every $Y (X)$. In this paper, we show that universal right-stability spaces are just Hilbert spaces; every injective space is universally left-stable; a Banach space $X$ isomorphic to a subspace of $\ell_\infty$ is universally left-stable if and only if it is isomorphic to $\ell_\infty$; and that a separable space $X$ satisfies the condition that $(X,Y)$ is left-stable for every separable $Y$ if and only if it is isomorphic to $c_0$.

math.FA↗