arXiv subjects

Cheick Toure

Publications and source records attributed to Cheick Toure.

3 recordsLinked to original sources

arXiv2025

The Multiplicative Kowalski-Slodkowski Theorem for Hermitian Algebras

We prove, for Hermitian algebras, the multiplicative version of the Kowalski-S\l{}odkowski Theorem which identifies the characters among the collection of all complex valued functions on a Banach algebra $A$ in terms of a spectral condition. Specifically, we show that, if $A$ is a Hermitian algebra, and if $\phi:A\mapsto\mathbb C$ is a continuous function satisfying $\phi(x)\phi(y) \in \sigma(xy)$ for all $x,y\in A$ (where $\sigma$ denotes the spectrum), then either $\phi$ or $-\phi$ is a character of $A$; of course the converse holds as well. Our proof depends fundamentally on the existence of positive elements and square roots in these algebras.

math.FA↗

arXiv2018

A Spectral Characterization of Isomorphisms on $C^\star$-Algebras

Following a result of Hatori, Miura and Tagaki ([4]) we give here a spectral characterization of an isomorphism from a $C^\star$-algebra onto a Banach algebra. We then use this result to show that a $C^\star$-algebra $A$ is isomorphic to a Banach algebra $B$ if and only if there exists a surjective function $ϕ:A\rightarrow B$ satisfying (i) $σ\left(ϕ(x)ϕ(y)ϕ(z)\right)=σ\left(xyz\right)$ for all $x,y,z\in A$ (where $σ$ denotes the spectrum), and (ii) $ϕ$ is continuous at $\mathbf 1$. A simple example shows that (i) cannot be relaxed to products of two elements, as is the case with commutative Banach algebras. Our results also elaborate on a paper ([3]) of Brešar and Špenko.

math.FA↗

arXiv2018

Truncation and Spectral Variation in Banach Algebras

Let $a$ and $b$ be elements of a semisimple, complex and unital Banach algebra $A$. Using subharmonic methods, we show that if the spectral containment $σ(ax)\subseteqσ(bx)$ holds for all $x\in A$, then $ax$ belongs to the bicommutant of $bx$ for all $x\in A$. Given the aforementioned spectral containment, the strong commutation property then allows one to derive, for a variety of scenarios, a precise connection between $a$ and $b$. The current paper gives another perspective on the implications of the above spectral containment which was also studied, not long ago, by J. Alaminos, M. Brešar et. al.

math.FA↗