The classical monotone convergence theorem of Beppo Levi fails in noncommutative $L_2$-spaces

Abstract:

Let $H$ be a complex Hilbert space and let $\mathfrak {A}$ be a von Neumann algebra over $H$ equipped with a faithful, normal state $\phi$. Then $\mathfrak {A}$ is a prehilbert space with respect to the inner product $\langle A\mid B\rangle := \phi (B^* A)$, whose completion $L_2 = L_2 (\mathfrak {A} ,\phi )$ is given by the Gelfand–Naimark–Segal representation theorem, according to which there exist a one-to-one $*$-homomorphism $\pi$ of $\mathfrak {A}$ into the algebra $\mathcal {L} (L_2)$ of all bounded linear operators acting on $L_2$ and a cyclic, separating vector $\omega \in L_2$ such that $\phi (A) = (\pi (A) \omega \mid \omega )$ for all $A\in \mathfrak {A}$. Given any separable Hilbert space $H$, we construct a faithful, normal state $\phi$ on $\mathcal {L} (H)$ and an increasing sequence $(A_n : n\ge 1)$ of positive operators acting on $H$ such that $(\phi (A^2_n) : n\ge 1)$ is bounded, but $(\pi (A_n) \omega : n\ge 1)$ fails to converge both bundlewise and in $L_2$-norm. We also present an example of an increasing sequence of positive operators which has a subsequence converging both bundlewise and in $L_2$-norm, but the whole sequence fails to converge in either sense. Finally, we observe that our results are linked to a previous one by R. V. Kadison.