A characterization of absolutely dilatable Schur multipliers
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912217967886336 |
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| author | Duquet, Charles Merdy, Christian Le |
| author_facet | Duquet, Charles Merdy, Christian Le |
| contents | Let $M$ be a von Neumann algebra equipped with a normal semi-finite faithful trace (nsf trace in short) and let $T\colon M\to M$ be a contraction. We say that $T$ is absolutely dilatable if there exist another von Neumann algebra $M'$ equipped with a nsf trace, a $w^*$-continuous trace preserving unital $*$-homomorphim $J\colon M\to M'$ and a trace preserving $*$-automomorphim $U\colon M'\to M'$ such that $T^k=E U^k J$ for all integer $k\geq 0$, where $E\colon M'\to M$ is the conditional expectation associated with $J$. Given a $σ$-finite measure space $(Ω,μ)$, we characterize bounded Schur multipliers $ϕ\in L^\infty(Ω^2)$ such that the Schur multiplication operator $T_ϕ\colon B(L^2(Ω))\to B(L^2(Ω))$ is absolutely dilatable. In the separable case, they are characterized by the existence of a von Neumann algebra $N$ with a separable predual, equipped with a normalized normal faithful trace $τ_N$, and of a $w^*$-continuous essentially bounded function $d\colonΩ\to N$ such that $ϕ(s,t)=τ_N(d(s)^*d(t))$ for almost every $(s,t)\inΩ^2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_08436 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A characterization of absolutely dilatable Schur multipliers Duquet, Charles Merdy, Christian Le Operator Algebras Functional Analysis 47A20, 46L06, 46E40 Let $M$ be a von Neumann algebra equipped with a normal semi-finite faithful trace (nsf trace in short) and let $T\colon M\to M$ be a contraction. We say that $T$ is absolutely dilatable if there exist another von Neumann algebra $M'$ equipped with a nsf trace, a $w^*$-continuous trace preserving unital $*$-homomorphim $J\colon M\to M'$ and a trace preserving $*$-automomorphim $U\colon M'\to M'$ such that $T^k=E U^k J$ for all integer $k\geq 0$, where $E\colon M'\to M$ is the conditional expectation associated with $J$. Given a $σ$-finite measure space $(Ω,μ)$, we characterize bounded Schur multipliers $ϕ\in L^\infty(Ω^2)$ such that the Schur multiplication operator $T_ϕ\colon B(L^2(Ω))\to B(L^2(Ω))$ is absolutely dilatable. In the separable case, they are characterized by the existence of a von Neumann algebra $N$ with a separable predual, equipped with a normalized normal faithful trace $τ_N$, and of a $w^*$-continuous essentially bounded function $d\colonΩ\to N$ such that $ϕ(s,t)=τ_N(d(s)^*d(t))$ for almost every $(s,t)\inΩ^2$. |
| title | A characterization of absolutely dilatable Schur multipliers |
| topic | Operator Algebras Functional Analysis 47A20, 46L06, 46E40 |
| url | https://arxiv.org/abs/2303.08436 |