Noncommutative BKW-Operators

TL;DR

This paper introduces and studies noncommutative BKW-operators within $C^*$-algebras, exploring their properties, connections to CP-extensions, and related concepts like hyperrigidity, extending classical approximation theorems into the noncommutative setting.

Contribution

It defines noncommutative BKW-operators, investigates their properties, links to CP-extensions, and extends classical Korovkin theorems to noncommutative operator frameworks.

Findings

Established the concept of noncommutative BKW-operators.

Connected noncommutative BKW-operators with unique CP-extensions.

Extended Korovkin-type theorems to noncommutative operator algebras.

Abstract

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 $\phi: A\rightarrow B(H)$ 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 $\phi_n:A\rightarrow B(H)$, $n=1,2,...,$ $\lim_{n\rightarrow \infty}\phi_n(s)=\phi(s),\forall ~s\in S$ in norm (or WOT or SOT) $\Rightarrow \lim_{n\rightarrow \infty}\phi_n(a)=\phi(a), \forall ~a\in A$ 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.