Bundle convergence of weighted sums of operators in noncommutative \(L_2\)-spaces (Q2777727)

Let \({\mathbf A}\) be a von Neumann algebra, acting on a Hilbert space \(H\), \(\varphi\) a faithful normal state on \({\mathbf A}\). The completion of \(({\mathbf A},\langle\cdot,\cdot\rangle)\), where \(\langle A,B\rangle:= \varphi(B^*A)\) is \(L_2= L_2({\mathbf A},\varphi)\). There are \(1-1^*\)-homomorphism \(\pi: {\mathbf A}\to B(L_2)\) and a cyclic and separating vector \(\omega\) such that \(\pi(A)= (\pi(A)\omega,\omega)\), \(A\in{\mathbf A}\).NEWLINENEWLINENEWLINEThe bundle convergence in \(L_2\) [\textit{E. Hensz}, \textit{R. Jajte} and \textit{A. Paszkiewicz}, Stud. Math. 120, No. 1, 23-46 (1996; Zbl 0856.46033)]: Given a sequence \((D_k)\subset{\mathbf A}\) such that \(\sum^\infty_{k=1} \varphi(D_k)< \infty\), the bundle \({\mathbf P}={\mathbf P}(D_k)L= \{P\in \text{Proj }{\mathbf A}: \sup_{n\geq 1}\|P(\sum^n_{k=1} D_k) P\|_\infty\) is finite and \(\|PD_nP\|_\infty\to 0\), \(n\to\infty\}\).NEWLINENEWLINENEWLINEA sequence \((\xi_n)\subset L_2\) is said to be bundle convergent to zero \(0\in L_2\), \(\xi_n@> b>> 0\), if there exists a sequence \((A_n)\subset{\mathbf A}\), such that \(\sum^\infty_{n=1}\|\xi_n- \pi(A_n)\omega\|^2< \infty\) and there exists a bundle \({\mathbf P}\) such that \(\forall P\in{\mathbf P}\|A_n P\|_\infty\to 0\), \(n\to\infty\).NEWLINENEWLINENEWLINETheorem 1. Let \((B_k)\) be a sequence of operators in \({\mathbf A}\) and \((a_k)\in\ell_2\). If \(A_n:= \sum^n_{k=1} a_k B_k\), \(n\in\mathbb{N}\), then \(\exists\xi\in L_2\), such that \(\|\pi(A_n)\omega- \xi\|\to 0\) and \(\pi(A_n)\omega@> b>>\xi\), \(n\to\infty\).NEWLINENEWLINENEWLINEThere are some results on noncommutative versions of the Borel-Cantelli lemmas. There is an example showing that a noncommutative version of the first Borel-Cantelli lemma in terms of bundle convergence does not exist. A noncommutative version of the second Borel-Cantelli lemma in terms of bundle convergence is proven. In the special case of \(\varphi\)-independent projections a noncommutative version of the first Borel-Cantelli lemma is derived. There are two problems.