In Caspers et al. (Can. J. Math. 75[6] [2022], 1-18), transference results between multilinear Fourier and Schur multipliers on noncommutative Lp-spaces were shown for unimodular groups. We propose a suitable extension of the definition of multilinear Fourier multipliers for non-unimodular groups and show that the aforementioned transference results also hold in this more general setting.

This thesis deals with multipliers and transference on noncommutative Lp-spaces. It falls within the realm of noncommutative harmonic analysis, i.e. harmonic analysis on noncommutative spaces. Such spaces appear in several areas of mathematics, such as non-abelian groups, quantum groups, noncommutative geometry and the general theory of operator algebras. Already the pioneering work of von Neumann, Gelfand and Dixmier showed that there are close connections between operator theory and representation theory of locally groups. These results formed the foundations for the theory of noncommutative harmonic analysis.....

We introduce the relative Haagerup approximation property for a unital, expected inclusion of arbitrary von Neumann algebras and show that if the smaller algebra is finite then the notion only depends on the inclusion itself, and not on the choice of the conditional expectation. Several variations of the definition are shown to be equivalent in this case, and in particular the approximating maps can be chosen to be unital and preserving the reference state. The concept is then applied to amalgamated free products of von Neumann algebras and used to deduce that the standard Haagerup property for a von Neumann algebra is stable under taking free products with amalgamation over finite-dimensional subalgebras. The general results are illustrated by examples coming from q-deformed Hecke-von Neumann algebras and von Neumann algebras of quantum orthogonal groups.

We consider semigroup BMO spaces associated with an arbitrary σ-finite von Neumann algebra (M, ϕ). We prove that BMO always admits a predual, extending results from the finite case. Consequently, we can prove—in the current setting of BMO—that they are Banach spaces and they interpolate with Lp as in the commutative situation, namely [BMO(M), L^◦_p(M)]_1/q ≈ L^◦_pq(M). We then study a new class of examples. We introduce the notion of Fourier–Schur multiplier on a compact quantum group and show that such multipliers naturally exist for SUq(2).

Let G be a locally compact unimodular group, and let φ be some function of n variables on G. To such a φ, one can associate a multilinear Fourier multiplier, which acts on some n-fold product of the noncommutative L _p-spaces of the group von Neumann algebra. One may also define an associated Schur multiplier, which acts on an n-fold product of Schatten classes S _p(L ₂(G). We generalize well-known transference results from the linear case to the multilinear case. In particular, we show that the so-called multiplicatively bounded (p1,....,pn)-norm"of a multilinear Schur multiplier is bounded above by the corresponding multiplicatively bounded norm of the Fourier multiplier, with equality whenever the group is amenable. Furthermore, we prove that the bilinear Hilbert transform is not bounded as a vector-valued map L _p1(R,S _p1) × L _p2 (R,S _p2), → L ₁(R,S ₁), whenever p1 and p2 are such that {equation presented}. A similar result holds for certain Calderón-Zygmund-type operators. This is in contrast to the nonvector-valued Euclidean case.