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Consider a completely bounded Fourier multiplier phi of a locally compact group G, and take 1 <= p <= infinity. One can associate to phi a Schur multiplier on the Schatten classes S_p(L^2 G), as well as a Fourier multiplier on Lp(LG), the…

Let G be a locally compact group L^p(G) be the usual L^p-space for 1 =< p =< infty and A(G) be the Fourier algebra of G. Our goal is to study, in a new abstract context, the completely bounded multipliers of A(G), which we denote…

In this work, we study Fourier multipliers on noncommutative spaces. In particluar, we show a simple proof of $L^p$-$L^q$ estimate of Fourier multipliers on general noncommutative spaces associated with semi-finite von Neumann algebras.…

The paper focuses on the behaviour of unimodular Fourier multipliers with exponential growth in the context of weighted $L^p$-spaces. Our main result shows that much of the general theory of multipliers is approachable through the theory of…

A Schur multiplier is a linear map on matrices which acts on its entries by multiplication with some function, called the symbol. We consider idempotent Schur multipliers, whose symbols are indicator functions of smooth Euclidean domains.…

For a locally compact group \(G\), let \(\mathcal{L}G\) denote its left group von Neumann algebra and let \(L^p(\mathcal{L}G)\), \(1 \le p \le \infty\), be the corresponding non-commutative \(L^p\)-space. Given \(\phi \in L^\infty(G)\), we…

In the article, we find new dilatation results on non-commutative $L_p$ spaces. We prove that any selfadjoint, unital, positive measurable Schur multiplier on some $B(L^2(\Sigma))$ admits, for all $1\leq p<\infty$, an invertible isometric…

Let $G$ be a locally compact unimodular group, and let $\phi$ be some function of $n$ variables on $G$. To such a $\phi$, one can associate a multilinear Fourier multiplier, which acts on some $n$-fold product of the non-commutative…

We study the algebra $\mathfrak{M}^{\infty,\mathrm{dec}}(G)$ of decomposable Fourier multipliers on the group von Neumann algebra $\mathrm{VN}(G)$ of a locally compact group $G$, and its relation to the Fourier-Stieltjes algebra…

We study BMO spaces associated with semigroup of operators and apply the results to boundedness of Fourier multipliers. We prove a universal interpolation theorem for BMO spaces and prove the boundedness of a class of Fourier multipliers on…

Let $E$ be an operator space in the sense of the theory recently developed by Blecher-Paulsen and Effros-Ruan. We introduce a notion of $E$-valued non commutative $L_p$-space for $1 \leq p < \infty$ and we prove that the resulting operator…

We establish an unbounded version of Stinespring's Theorem and a lifting result for Stinespring representations of completely positive modular maps defined on the space of all compact operators. We apply these results to study positivity…

We show that the multiplier algebra of the Fourier algebra on a locally compact group $G$ can be isometrically represented on a direct sum on non-commutative $L^p$ spaces associated to the right von Neumann algebra of $G$. If these spaces…

We study Fourier multiplier operators associated with symbols $\xi\mapsto \exp(i\lambda\phi(\xi/|\xi|))$, where $\lambda$ is a real number and $\phi$ is a real-valued $C^\infty$ function on the standard unit sphere…

In this paper we have studied Fourier multipliers and Littlewood-Paley square functions in the context of modulation spaces. We have also proved that any bounded linear operator from modulation space $\mathcal{M}_{p,q}(\R^n), 1\leq p,q\leq…

We introduce the notion of a regular mapping on a non-commutative $L_p$-space associated to a hyperfinite von Neumann algebra for $1\le p\le \infty$. This is a non-commutative generalization of the notion of regular or order bounded map on…

We describe the idempotent Fourier multipliers that act contractively on $H^p$ spaces of the $d$-dimensional torus $\mathbb{T}^d$ for $d\geq 1$ and $1\leq p \leq \infty$. When $p$ is not an even integer, such multipliers are just…

The notion of invariant operators, or Fourier multipliers, is discussed for densely defined operators on Hilbert spaces, with respect to a fixed partition of the space into a direct sum of finite dimensional subspaces. As a consequence,…

In this paper we establish the $L^p$-$L^q$ boundedness of Fourier multipliers on locally compact separable unimodular groups for the range of indices $1<p\leq 2 \leq q<\infty$. Our approach is based on the operator algebras techniques. The…

In arXiv:2206.00549, transference results between multilinear Fourier and Schur multipliers on noncommutative $L_p$-spaces were shown for unimodular groups. We propose a suitable extension of the definition of multilinear Fourier…