Cesàro summability of subsequence of the partial sums of Fourier series in operator-valued setting

Abstract

Let $f\in L_1(L_\infty ({\mathbb {T}})\bar{\otimes }{\mathcal {M}})$, where ${\mathbb {T}}$ is the classical torus, ${\mathcal {M}}$ is a semi-finite von Neumann algebra. We prove the noncommutative weak type maximal inequality

$$\begin{aligned} \Vert (\gamma _n(f))_n\Vert _{\Lambda _{1,\infty } ({\mathcal {N}},\ell _{\infty })}\le C\Vert f\Vert _{L_1({\mathcal {N}})}, \end{aligned}$$

where $\gamma _n(f)$ is the Cesàro means of the subsequence of the partial sums sequence $(S_n(f))_{n\ge 0}$. As a consequence, the Cesàro means $\gamma _n(f)$ converges bilaterally almost uniformly to f whenever $n\rightarrow \infty$.