Noncommutative BKW-Operators
Arunkumar C. S., Sruthymurali
TL;DR
This work develops a noncommutative BKW-operator framework for unital $C^*$-algebras via operator systems and unital completely positive maps, linking BKW behavior to unique CP-extensions (Theorem A) and introducing $\eta$-hyperrigidity with equivalent CP-extension characterizations (Theorem B). It proves a noncommutative Korovkin-type theorem extending Popa's result to Schwarz and $2$-positive maps (with a CP-map corollary), including concrete examples such as $\phi(x)=\lambda T^*xT$ and a normalization to a *-homomorphism case. The paper also establishes existence of noncommutative BKW-operators for any operator system and discusses the role of boundary representations, outlining open questions on nonseparable extensions and the uniqueness of Arveson extensions along with future directions connecting BKW-operators to hyperrigidity. These results advance noncommutative approximation theory within the operator algebra framework and suggest practical implications for CP-extension problems and related numerical methods.
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 $φ: A\rightarrow B(H)$ 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.