arXiv subjects

Osamu Hatori

Publications and source records attributed to Osamu Hatori.

22 records · Page 2Linked to original sources

arXiv2009

An algebraic property of an isometry between the groups of invertible elements in Banach algebras

We show that if $T$ is an isometry (as metric spaces) between the invertible groups of unital Banach algebras, then $T$ is extended to a surjective real-linear isometry up to translation between the two Banach algebras. Furthermore if the underling algebras are closed unital standard operator algebras, $(T(e_A))^{-1}T$ is extended to a surjective real algebra isomorphism; if $T$ is a surjective isometry from the invertible group of a unital commutative Banach algebra onto that of a unital semisimple Banach algebra, then $(T(e_A))^{-1}T$ is extended to a surjective isometrical real algebra isomorphism between the two underling algebras.

math.FA↗

arXiv2009

Polynomially spectrum-preserving maps between commutative Banach algebras

Let $A$ and $B$ be unital semi-simple commutative Banach algebras. In this paper we study two-variable polynomials $p$ which satisfy the following property: a map $T$ from $A$ onto $B$ such that the equality \[ σ(p(Tf,Tg))=σ(p(f,g)), \quad f,g \in A \] holds is an algebra isomorphism.

math.FA↗

arXiv2009

Isometries between groups of invertible elements in Banach algebras

We show that if $T$ is an isometry (as metric spaces) from an open subgroup of the group of the invertible elements in a unital semisimple commutative Banach algebra onto an open subgroup of the group of the invertible elements in a unital Banach algebra, then $T(1)^{-1}T$ is an isometrical group isomorphism. In particular, $T(1)^{-1}T$ is extended to an isometrical real algebra isomorphism from $A$ onto $B$.

math.FA↗

arXiv2009

Multiplicatively spectrum-preserving and norm-preserving maps between invertible groups of commutative Banach algebras

Let $A$ and $B$ be unital semisimple commutative Banach algebras and $T$ a map from the invertible group $A^{-1}$ onto $B^{-1}$. Linearity and multiplicativity of the map are not assumed. We consider the hypotheses on $T$: (1) $σ(TfTg)=σ(fg)$; (2) $σ_π(TfTg-α)\cap σ_π(fg-α)\ne \emptyset$; (3) $\mathrm{r} (TfTg-α)=\mathrm{r}(fg-α)$ hold for some non-zero complex number $α$ and for every $f, g\in A^{-1}$, where $σ(\cdot)$ (resp. $σ_π(\cdot)$) denotes the (resp. peripheral) spectrum and $\rr(\cdot)$ denotes the spectral radius. Under each of the hypotheses we show representations for $T$ and under additional assumptions we show that $T$ is extended to an algebra isomorphism. In particular, if $T$ is a surjective group homomorphism such that $T$ preserves the spectrum or $T$ is a surjective isometry with respect to the spectral radius, then $T$ is extended to an algebra isomorphism. Similar results holds for maps from $A$ onto $B$.

math.FA↗