Multipliers for noncommutative Walsh-Fourier series

Abstract:

We consider multipliers for noncommutative Walsh-Fourier series. Let $\mathcal {R}$ be the type $II_1$ hyperfinite factor. For $x \in L_1(\mathcal {R})$, $0<\alpha <1$, the multiplier transformation of $x$ is defined by setting \[ J^\alpha x = \sum _{\gamma \in \mathcal {F}}2^{-\gamma (1)\alpha }\hat {x}(\gamma )w_\gamma ,\] where $(w_\gamma )_{\gamma \in \mathcal {F}}$ is the noncommutative Walsh system in $\mathcal {R}$ and $\sum _{\gamma \in \mathcal {F}}\hat {x}(\gamma )w_\gamma$ is the Walsh-Fourier series of $x$. It is shown that \[ \|J^\alpha x\|_{L_{1/(1-\alpha ),\infty }(\mathcal {R})}\leq c \|x\|_{L_1(\mathcal {R})},\] where $c$ is a constant depending only on $\alpha$. Via interpolations, we deduce that $J^\alpha$ is bounded from $L_{p}(\mathcal {R})$ into $L_{q}(\mathcal {R})$ where $1<p<q$ and $\alpha =1/p-1/q$, thus providing a noncommutative analogue of a classical result of Watari.