Noncommutative BKW-Operators
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914118699581440 |
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| author | S., Arunkumar C. Sruthymurali |
| author_facet | S., Arunkumar C. Sruthymurali |
| contents | Inspired by the classical Bohman-Korovkin-Wulbert (BKW) operators, we initiate a study of noncommutative BKW-operators. Let $A$ be a unital $C^*$-algebra, and $S$ be a set of generators of $A$. A unital completely positive (UCP)-map $ϕ: A\rightarrow B(H)$ is said to be a \textit{noncommutative BKW-operator} for $S$ with respect to norm or weak operator topology (WOT) or strong operator topology (SOT) if for any sequence of UCP-maps $ϕ_n:A\rightarrow B(H)$, $n=1,2,...,$ $\lim_{n\rightarrow \infty}ϕ_n(s)=ϕ(s),\forall ~s\in S$ in norm (or WOT or SOT) $\Rightarrow \lim_{n\rightarrow \infty}ϕ_n(a)=ϕ(a), \forall ~a\in A$ in norm (or WOT or SOT, respectively). We identify a connection between noncommutative BKW-operators and the unique CP-extension of UCP-maps. We have discussed several examples and explored different notions of noncommutative BKW-operators and their interconnections. Additionally, we introduce the concept of hyperrigidity with respect to a UCP-map and characterize it along the lines of Arveson. Although independent yet related to noncommutative BKW-operators, we provide a noncommutative version of operator version of the Korovkin theorem recently proposed by D. Popa. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_24470 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Noncommutative BKW-Operators S., Arunkumar C. Sruthymurali Operator Algebras Functional Analysis 41A36, 46L05 Inspired by the classical Bohman-Korovkin-Wulbert (BKW) operators, we initiate a study of noncommutative BKW-operators. Let $A$ be a unital $C^*$-algebra, and $S$ be a set of generators of $A$. A unital completely positive (UCP)-map $ϕ: A\rightarrow B(H)$ is said to be a \textit{noncommutative BKW-operator} for $S$ with respect to norm or weak operator topology (WOT) or strong operator topology (SOT) if for any sequence of UCP-maps $ϕ_n:A\rightarrow B(H)$, $n=1,2,...,$ $\lim_{n\rightarrow \infty}ϕ_n(s)=ϕ(s),\forall ~s\in S$ in norm (or WOT or SOT) $\Rightarrow \lim_{n\rightarrow \infty}ϕ_n(a)=ϕ(a), \forall ~a\in A$ in norm (or WOT or SOT, respectively). We identify a connection between noncommutative BKW-operators and the unique CP-extension of UCP-maps. We have discussed several examples and explored different notions of noncommutative BKW-operators and their interconnections. Additionally, we introduce the concept of hyperrigidity with respect to a UCP-map and characterize it along the lines of Arveson. Although independent yet related to noncommutative BKW-operators, we provide a noncommutative version of operator version of the Korovkin theorem recently proposed by D. Popa. |
| title | Noncommutative BKW-Operators |
| topic | Operator Algebras Functional Analysis 41A36, 46L05 |
| url | https://arxiv.org/abs/2510.24470 |