Multipliers of Beurling-Fourier algebras

Abstract: For a locally compact group G we introduce and study the reduced Beurling-Fourier-Stieltjes algebra, a weighted analogue of the reduced Fourier-Stieltjes algebra, together with the algebra of completely bounded multipliers of the associated weighted Fourier algebra. We show, in particular, that these two algebras coincide when G is amenable. For a general locally compact group G, we identify them as subspaces of the reduced Fourier-Stieltjes algebra and of the space of functions that locally belong to the Fourier algebra, respectively. Furthermore, we establish sufficient conditions on the group and the weight under which the algebra of completely bounded multipliers of the weighted Fourier algebra embeds into its unweighted counterpart.

• The paper establishes a complete characterization of multiplier algebras for Beurling-Fourier algebras in both amenable and non-amenable groups.
• It extends the classical Herz-Schur multiplier theory to the weighted setting by introducing the reduced Beurling-Fourier-Stieltjes algebra.
• The work explores weight lifting techniques and tensor product structures, providing new insights into noncommutative harmonic analysis.

Multipliers of Beurling-Fourier Algebras: Structural and Multiplier Theory

Introduction and Main Contributions

The article "Multipliers of Beurling-Fourier algebras" (2606.06675) provides a comprehensive operator-theoretic analysis of weighted analogues of Fourier and Fourier-Stieltjes algebras on locally compact groups—specifically, Beurling-Fourier algebras and their reduced Fourier-Stieltjes counterparts. The authors develop a general theory of the algebraic and operator space structures of these algebras, introduce the reduced Beurling-Fourier-Stieltjes algebra $B_r(G, \omega)$ as a weighted version of $B_r(G)$, and provide a complete description of their multiplier algebras (in particular, the space of completely bounded multipliers) for both amenable and non-amenable groups.

A central achievement is the identification of precise relationships between the weighted Fourier algebras, their (reduced) Fourier-Stieltjes analogues, and the corresponding spaces of (completely bounded) multipliers. The results generalize classical connections, notably the Herz-Schur description of completely bounded multipliers, to the weighted setting. The analysis includes an exploration of when weighted multiplier algebras embed into their unweighted analogues, leading to new insights on the structure of both commutative and noncommutative harmonic analysis.

Weighted Fourier and Fourier-Stieltjes Algebras

The Eymard Fourier algebra $A(G)$ is the unique predual of the group von Neumann algebra $VN(G)$ and central to noncommutative harmonic analysis. The Beurling-Fourier algebra $A(G, \omega)$, as developed in [gllst, gt, LeSa, lst], modifies $A(G)$ by introducing an operator weight $\omega$ (a "weight inverse" in the sense of [ors]) belonging to the multiplier algebra of the reduced group $C^*$-algebra $M(C^*_r(G))$.

Key properties are established, including:

• $A(G, \omega)$ is a commutative Banach algebra, completely contractive with respect to an operator space structure, and its Gelfand spectrum embeds $B_r(G)$0.
• The spectrum, norm structure, and imbeddings of $B_r(G)$1 significantly depend on the properties of $B_r(G)$2 and $B_r(G)$3.

The reduced Beurling-Fourier-Stieltjes algebra $B_r(G)$4 is then defined as $B_r(G)$5, offering a dual Banach algebra structure and generalizing the classical $B_r(G)$6. The inclusion $B_r(G)$7 is always proper unless $B_r(G)$8 is invertible.

Multiplier Structures and Completely Bounded Multipliers

Main Theorems

The article establishes the following core structural facts:

• Faithfulness and Ideal Property: $B_r(G)$9 is always an ideal in $A(G)$0, and $A(G)$1 is a faithful dual Banach algebra.
• Reduction to Unweighted Case in Amenability: $A(G)$2 coincides with the space of completely bounded multipliers $A(G)$3 if and only if $A(G)$4 is amenable. Otherwise, the inclusion $A(G)$5 is proper.
• Weighted Herz-Schur Multipliers: $A(G)$6 is concretely characterized: any $A(G)$7 admits a representation as a (possibly weighted) Schur multiplier, extending the Herz-Schur description to the weighted setting.
• Embedding into Locally Defined Fourier Algebras: For general groups, $A(G)$8 embeds into $A(G)$9, the algebra of functions locally in the Fourier algebra.

Furthermore, the authors show that for central weights or weights induced from normal amenable subgroups, $VN(G)$0 embeds into the unweighted completely bounded multipliers, with further sufficient conditions linking this embedding to the existence of traces or suitable convex hull properties in the multiplier algebra of $VN(G)$1.

Contradictory and Strong Structural Claims

The following sharp dichotomy is rigorously demonstrated:

• Amenable Case: $VN(G)$2 if and only if $VN(G)$3 is amenable.
• Non-amenable Case: $VN(G)$4 is properly contained in $VN(G)$5; i.e., for non-amenable groups, there are more completely bounded multipliers than elements in $VN(G)$6.

The paper also presents examples of weights and group structures for which neither $VN(G)$7 nor $VN(G)$8 exhaust $VN(G)$9. For connected, non-compact, finite-center semisimple Lie groups, it is shown that the requisite convex orbit conditions are never satisfied, precluding embeddings into the unweighted multiplier algebra for arbitrary weights.

Weight Lifting and Structure of Examples

A substantial part of the work analyzes lifting weights from subgroups via $A(G, \omega)$0-homomorphisms between group von Neumann algebras, generalizing the Herz restriction theorem to the Beurling setting. This provides concrete families of non-invertible weights whenever $A(G, \omega)$1 contains a closed subgroup isomorphic to $A(G, \omega)$2 or $A(G, \omega)$3, significantly broadening the landscape of available weighted Fourier algebras.

By careful consideration of tensor products and operator space structures, it is shown that:

• $A(G, \omega)$4 is completely isometrically isomorphic to $A(G, \omega)$5.
• $A(G, \omega)$6 is always a dual Banach algebra, and the space spanned by $A(G, \omega)$7, $A(G, \omega)$8, is norm dense in $A(G, \omega)$9.

The duality structure and operator module tensor product techniques are extended from classical Fourier algebras to the weighted setting.

Implications and Future Directions

The algebraic and operator algebraic characterizations developed in this work have several theoretical and practical implications:

• They provide a robust framework for studying weighted approximation properties, duality, and spectral theory in abstract and noncommutative harmonic analysis.
• The extension of Herz-Schur multiplier theory introduces new tools to study completely bounded multipliers in contexts relevant to representation theory, $A(G)$0-algebras, and quantum groups.
• The characterization of when $A(G)$1 embeds into $A(G)$2 informs future investigations of exactness, approximation, and amenability phenomena in both group and quantum group settings.
• The results enable further study of analytic subalgebras and spectra of Beurling-Fourier algebras, especially in settings where weights arise from geometric or representation-theoretic data.

Potential directions include systematic analysis of weighted Fourier algebras on non-amenable groups, more general quantum groups, and connections to noncommutative geometric analysis of operator algebras.

Conclusion

This article establishes a comprehensive theory of multipliers for Beurling-Fourier algebras, introducing and analyzing the reduced Beurling-Fourier-Stieltjes algebra and providing deep, precise results on the structure of completely bounded multipliers. The work clarifies the role of amenability, central weights, and group structure in the embedding problems for weighted multiplier algebras, significantly advancing both abstract harmonic analysis and the theory of operator algebras associated with locally compact groups.