Algebra isomorphisms between standard operator algebras

Thomas Tonev, and Aaron Luttman. "Algebra isomorphisms between standard operator algebras." Studia Mathematica 191.2 (2009): 163-170. <http://eudml.org/doc/286573>.

@article{ThomasTonev2009,
abstract = {If X and Y are Banach spaces, then subalgebras ⊂ B(X) and ⊂ B(Y), not necessarily unital nor complete, are called standard operator algebras if they contain all finite rank operators on X and Y respectively. The peripheral spectrum of A ∈ is the set $σ_\{π\}(A) = \{λ ∈ σ(A): |λ| = max_\{z∈σ(A)\} |z|\}$ of spectral values of A of maximum modulus, and a map φ: → is called peripherally-multiplicative if it satisfies the equation $σ_\{π\}(φ(A)∘φ(B)) = σ_\{π\}(AB)$ for all A,B ∈ . We show that any peripherally-multiplicative and surjective map φ: → , neither assumed to be linear nor continuous, is a bijective bounded linear operator such that either φ or -φ is multiplicative or anti-multiplicative. This holds in particular for the algebras of finite rank operators or of compact operators on X and Y and extends earlier results of Molnár. If, in addition, $σ_\{π\}(φ(A₀)) ≠ -σ_\{π\}(A₀)$ for some A₀ ∈ then φ is either multiplicative, in which case X is isomorphic to Y, or anti-multiplicative, in which case X is isomorphic to Y*. Therefore, if X ≇ Y* then φ is multiplicative, hence an algebra isomorphism, while if X ≇ Y, then φ is anti-multiplicative, hence an algebra anti-isomorphism.},
author = {Thomas Tonev, Aaron Luttman},
journal = {Studia Mathematica},
keywords = {standard operator algebra; spectrum; peripheral spectrum; peripherally-multiplicative map; algebra isomorphism},
language = {eng},
number = {2},
pages = {163-170},
title = {Algebra isomorphisms between standard operator algebras},
url = {http://eudml.org/doc/286573},
volume = {191},
year = {2009},
}

TY - JOUR
AU - Thomas Tonev
AU - Aaron Luttman
TI - Algebra isomorphisms between standard operator algebras
JO - Studia Mathematica
PY - 2009
VL - 191
IS - 2
SP - 163
EP - 170
AB - If X and Y are Banach spaces, then subalgebras ⊂ B(X) and ⊂ B(Y), not necessarily unital nor complete, are called standard operator algebras if they contain all finite rank operators on X and Y respectively. The peripheral spectrum of A ∈ is the set $σ_{π}(A) = {λ ∈ σ(A): |λ| = max_{z∈σ(A)} |z|}$ of spectral values of A of maximum modulus, and a map φ: → is called peripherally-multiplicative if it satisfies the equation $σ_{π}(φ(A)∘φ(B)) = σ_{π}(AB)$ for all A,B ∈ . We show that any peripherally-multiplicative and surjective map φ: → , neither assumed to be linear nor continuous, is a bijective bounded linear operator such that either φ or -φ is multiplicative or anti-multiplicative. This holds in particular for the algebras of finite rank operators or of compact operators on X and Y and extends earlier results of Molnár. If, in addition, $σ_{π}(φ(A₀)) ≠ -σ_{π}(A₀)$ for some A₀ ∈ then φ is either multiplicative, in which case X is isomorphic to Y, or anti-multiplicative, in which case X is isomorphic to Y*. Therefore, if X ≇ Y* then φ is multiplicative, hence an algebra isomorphism, while if X ≇ Y, then φ is anti-multiplicative, hence an algebra anti-isomorphism.
LA - eng
KW - standard operator algebra; spectrum; peripheral spectrum; peripherally-multiplicative map; algebra isomorphism
UR - http://eudml.org/doc/286573
ER -