arXiv subjects

Curt Healey

Publications and source records attributed to Curt Healey.

3 recordsLinked to original sources

arXiv2025

On a comprehensive review of a proof of L\"owner's theorem

Recent studies in Kubo-Ando theory make frequent use of the relationship between Kubo-Ando connections and positive operator monotone functions. This relationship is deeply connected to L\"owner's theorem and our aim is to provide a comprehensive review of one of the proofs of L\"owner's theorem. Our motivation arises from the fact that the foundational components upon which the theorem rests are found within a variety of sources, rendering it difficult to obtain a complete understanding of the proof without engaging in substantial external consultation. By consolidating these elements into a single, continuous account, the proof becomes substantially more accessible and may be assimilated with greater clarity and efficiency.

math.FA↗

arXiv2024

Extending surjective maps preserving the norm of symmetric kubo-ando means

Recently, the question of whether surjective maps preserving the norm of a symmetric Kubo-Ando mean can be extended to Jordan $\ast$-isomorphisms has been tackled. The question was affirmatively answered for surjective maps between $C^{*}$-algebras for certain specific classes of symmetric Kubo-Ando means. Here, we give a comprehensive answer to this question for surjective maps between $AW^{*}$-algebras preserving the norm of any symmetric Kubo-Ando mean.

math.OA↗

arXiv2023

Every symmetric Kubo-Ando connection has the order-determining property on $\mathcal B(H)$

In \cite{molnar} L.~Molnar studied the question of whether the L\"owner partial order on the positive cone of an operator algebra is determined by the norm of any arbitrary Kubo-Ando mean. He affirmatively answered the question for certain classes of Kubo-Ando means and left as an open problem the general case. We here give an answer to this question, by showing that the norm of every symmetric Kubo-Ando mean $\sigma$ on $\mathcal B(H)$ is order-determining, i.e. if $A, B\in \mathcal B(H)^{{\sss{++}}}$ satisfy $\Vert A\sigma X\Vert \le \Vert B\sigma X\Vert$ for every $X\in \mathcal B(H)^{\sss{++}}$, then $A\le B$.

math.FA↗