Homomorphisms of Fourier algebras and transference results
M. Anoussis, G. K. Eleftherakis, A. Katavolos
备注：
This preprint will be restructured and will be replaced by `Synthetic properties of locally compact groups: preservation and transference' (submitted to arXiv) and `Homomorphisms of Fourier algebras' (in preparation)
We prove that if $ρ: A(H) \to B(G)$ is a homomorphism between the Fourier algebra of a locally compact group $H$ and the Fourier-Stieltjes algebra of a locally compact group $G$ induced by a mixed piecewise affine map $α: G \to H$, then $ρ$ extends to a w*-w* continuous map between the corresponding $L^\infty$ algebras if and only if $α$ is an open map. Using techniques from TRO equivalence of masa bimodules we prove various transference results: We show that when $α$ is a group homomorphism which pushes forward the Haar measure of $G$ to a measure absolutely continuous with respect to the Haar measure of $H$, then $(α\timesα)^{-1}$ preserves sets of compact operator synthesis, and conversely when $α$ is onto. We also prove similar preservation properties for operator Ditkin sets and operator M-sets, obtaining preservation properties for M-sets as corollaries. Some of these results extend or complement existing results of Ludwig, Shulman, Todorov and Turowska.
Using techniques from TRO equivalence of masa bimodules we prove various transference results: We show that when $α$ is a group homomorphism which pushes forward the Haar measure of $G$ to a measure absolutely continuous with respect to the Haar measure on $H$, then $(α\timesα)^{-1}$ preserves sets of compact operator synthesis, and conversely when $α$ is onto. We also prove similar preservation results for operator Ditkin sets and operator M-sets, obtaining preservation results for M-sets as corollaries. Some of these results extend or complement existing results of Ludwig, Shulman, Todorov a…
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 non-commutative $L_p$-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 $(p_1,\ldots,p_n)$-norm' of a multilinear Sch…
The standard proof of the equivalence of Fourier type on R^d and on the torus T^d is usually stated in terms of an implicit constant, defined as the minimum of a sum of powers of sinc functions. In this note we compute this minimum explicitly.
Every homomorphism $φ: B(G) \rightarrow B(H)$ between Fourier-Stieltjes algebras on locally compact groups $G$ and $H$ is determined by a continuous mapping $α: Y \rightarrow Δ(B(G))$, where $Y$ is a set in the open coset ring of $H$ and $Δ(B(G))$ is the Gelfand spectrum of $B(G)$ (a $*$-semigroup). We exhibit a large collection of maps $α$ for which $φ=j_α: B(G) \rightarrow B(H)$ is a completely positive/completely contractive/completely bounded homomorphism and establish converse statements in several instances. For example, we fully characterize all completely positive/completely contractiv…
We provide necessary and sufficient conditions for the existence of idempotents of arbitrarily large norms in the Fourier algebra A(G) and the Fourier-Stieltjes algebra B(G) of a locally compact group G. We prove that the existence of idempotents of arbitrarily large norm in B(G) implies the existence of homomorphisms of arbitrarily large norm from A(H) into B(G) for every locally compact group H. A partial converse is also obtained: the existence of homomorphisms of arbitrarily large norm from A(H) into B(G) for some amenable locally compact group H implies the existence of idempotents of arb…
We investigate conditions for the extendibility of continuous algebra homomorphisms $ϕ$ from the Fourier algebra $A(F)$ of a locally compact group $F$ to the Fourier-Stieltjes algebra $B(G)$ of a locally compact group $G$ to maps between the corresponding $L^\infty$ algebras which are weak* continuous. When $ϕ$ is completely bounded and $F$ is amenable, it is induced by a piecewise affine map $α: Y\to F$ where $Y\subseteq G$. We show that extendibility of $ϕ$ is equivalent to $α$ being an open map. We also study the dual problem for contractive homomorphisms $ϕ: L^1(F)\to M(G)$. We show that $…
In this paper we prove $L^p$ estimates for Stein's square functions associated to Fourier-Bessel expansions. Furthermore we prove transference results for square functions from Fourier-Bessel series to Hankel transforms. Actually, these are transference results for vector-valued multipliers from discrete to continuous in the Bessel setting. As a consequence, we deduce the sharpness of the range of $p$ for the $L^p$-boundedness of Fourier-Bessel Stein's square functions from the corresponding property for Hankel-Stein square functions. Finally, we deduce $L^p$ estimates for Fourier-Bessel multi…
Inspired by Kalton and Wood's work on group algebras, we describe almost completely contractive algebra homomorphisms from Fourier algebras into Fourier-Stieltjes algebras (endowed with their canonical operator space structure). We also prove that two locally compact groups are isomorphic if and only if there exists an algebra isomorphism $T$ between the associated Fourier algebras (resp. Fourier-Stieltjes algebras) with completely bounded norm $\| T \|_{cb} < \sqrt {3/2}$ (resp. $ \| T \|_{cb} < \sqrt {5}/2$). We show similar results involving the norm distortion $\| T \| \| T ^{-1} \|$ with …
In this paper, we follow two main goals. In the first attempt, we give some functorial properties of the $p$-analog of the Fourier-Stieltjes algebras in which we generalize some previously existed definitions and theorems in Arsac and Cowling's works, to utilize them to prove $p$-complete boundedness of some well-known maps on these algebras. In the second part, as an application of these generalizations, we prove $p$-completely boundedness of homomorphisms which are induced by continuous and proper piecewise affine maps that is a generalization of Ilie's work on Figà-Talamanca-Herz algebras.
In this article, we give a representation of bounded complex linear operators which preserve idempotent elements on the Fourier algebra of a locally compact group. When such an operator is moreover positive or contractive, we show that the operator is induced by either a continuous group homomorphism or a continuous group anti-homomorphism. If the groups are totally disconnected, bounded homomorphisms on the Fourier algebra can be realised by the idempotent preserving operators.
Let $A(G)$ and $B(H)$ be the Fourier and Fourier-Stieltjes algebras of locally compact groups $G$ and $H$, respectively. Ilie and Spronk have shown that continuous piecewise affine maps $α: Y \subseteq H\rightarrow G$ induce completely bounded homomorphisms $Φ:A(G)\rightarrow B(H)$, and that when $G$ is amenable, every completely bounded homomorphism arises in this way. This generalised work of Cohen in the abelian setting. We believe that there is a gap in a key lemma of the existing argument, which we do not see how to repair. We present here a different strategy to show the result, which in…
The aim of this paper is to show that any continuous *-homomorphism of L1(Cn)(with twisted convolution as multipli- cation) into B(L2(Rn)) is essentially a Weyl transform. From this we deduce a similar characterisation for the group Fourier transform on the Heisenberg group, in terms of convolution.
Let G be a locally compact group, and let A(G) and B(G) denote its Fourier and Fourier-Stieltjes algebras. These algebras are dual objects of the group and measure algebras, L^1(G) and M(G), in a sense which generalizes the Pontryagin duality theorem on abelian groups. We wish to consider the amenability properties of A(G) and B(G) and compare them to such properties for L^1(G) and M(G). For us, ``amenability properties'' refers to amenability, weak amenability, and biflatness, as well as some properties which are more suited to special settings, such as the hyper-Tauberian property for semisi…
We show that for a locally compact group $G$, amongst a class which contains amenable and small invariant neighbourhood groups, that its Fourier algebra $A(G)$ satisfies a completely bounded version Pisier's similarity property with similarity degree at most $2$. Specifically, any completely bounded homomorphism $π: A(G)\to B(H)$ admits an invertible $S$ in $B(H)$ for which $\|S\|\|S^{-1}\|\leq ||π||_{cb}^2$ and $S^{-1}π(\cdot)S$ extends to a $*$-representation of the $C^*$-algebra $C_0(G)$. This significantly improves some results due to Brannan and Samei (J. Funct. Anal. 259, 2010) and Brann…
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 are given their canonical Operator space structure, then we get a completely isometric representation of the completely bounded multiplier algebra. We make a careful study of the non-commutative $L^p$ spaces we construct, and show that they are completely isometric to those considered recently by Forrest, Lee and Samei; we improve a result about module homomo…
Let $H$ and $K$ be locally compact groups and also $τ:H\to Aut(K)$ be a continuous homomorphism and $G_τ=H\ltimes_τK$ be the semi-direct product of $H$ and $K$ with respect to the continuous homomorphism $τ$. This paper presents a novel approach to the Fourier analysis of $G_τ$, when $K$ is abelian. We define the $τ$-dual group $G_{\hatτ}$ of $G_τ$ as the semi-direct product $H\ltimes_{\hatτ}\hat{K}$, where $\hatτ:H\to Aut(\hat{K})$ defined via (\ref{A}). We prove a Ponterjagin duality Theorem and also we study $τ$-Fourier transforms on $G_τ$. As a concrete application we show that how these t…
Given a locally compact quantum group $\mathbb G$, we study the structure of completely bounded homomorphisms $π:L^1(\mathbb G)\rightarrow\mathcal B(H)$, and the question of when they are similar to $\ast$-homomorphisms. By analogy with the cocommutative case (representations of the Fourier algebra $A(G)$), we are led to consider the associated map $π^*:L^1_\sharp(\mathbb G) \rightarrow \mathcal B(H)$ given by $π^*(ω) = π(ω^\sharp)^*$. We show that the corepresentation $V_π$ of $L^\infty(\mathbb G)$ associated to $π$ is invertible if and only if both $π$ and $π^*$ are completely bounded. Moreo…
We establish precise regularity conditions for $L_p$-boundedness of Fourier multipliers in the group algebra of $SL_n(\mathbf{R})$. Our main result is inspired by Hörmander-Mikhlin criterion from classical harmonic analysis, although it is substantially and necessarily different. Locally, we get sharp growth rates of Lie derivatives around the singularity and nearly optimal regularity order. The asymptotics also match Mikhlin formula for a exponentially growing weight with respect to the word length. Additional decay comes imposed by this growth and Mikhlin condition for high order terms. Laff…
We study the geometry of infinitely presented groups satisfying the small cancelation condition C'(1/8), and define a standard decomposition (called the criss-cross decomposition) for the elements of such groups. We use it to prove the Rapid Decay property for groups with the stronger small cancelation property C'(1/10). As a consequence, the Metric Approximation Property holds for the reduced C*-algebra and for the Fourier algebra of such groups. Our method further implies that the kernel of the comparison map between the bounded and the usual group cohomology in degree 2 has a basis of power…
In this paper we solve three problems in noncommutative harmonic analysis which are related to endpoint inequalities for singular integrals. In first place, we prove that an $L_2$-form of Hörmander's kernel condition suffices for the weak type (1,1) of Calderón-Zygmund operators acting on matrix-valued functions. To that end, we introduce an improved CZ decomposition for martingale filtrations in von Neumann algebras, and apply a very simple unconventional argument which notably avoids pseudolocalization. In second place, we establish as well the weak $L_1$ endpoint for matrix-valued CZ operat…
In this work, we solve the problem explicitly stated at the end of a paper of Junge, Mei and Parcet [JEMS2018, Problem C.5] for a large class of groups including all amenable groups and free groups. More precisely, we prove that the Hodge-Dirac operator of the canonical "hidden" noncommutative geometry associated with a Markov semigroup $(T_t)_{t \geq 0}$ of Fourier multipliers is bisectorial and admits a bounded $\mathrm{H}^\infty$ functional calculus on a bisector which implies a positive answer to the quoted problem. Our result can be seen as a strengthening of the dimension free estimates …
This paper is devoted to studying certain topological properties of the maximal ideal space of the measure algebra on the circle group. In particular, we focus on Cech cohomologies of this space. Moreover, we show that the Gelfand space of $M(\mathbb{T})$ is not separable. On the other hand, we give a direct procedure to recover many copies of $β\mathbb{Z}$ in $\mathfrak{M}(M(\mathbb{T}))$, but we also show that this is result is not accessible in the most natural way (namely, by the canonical mapping induced by homomorphism assigning to measure its Fourier - Stieltjes transform).
Let ${\cal{M}}$ be a complete Riemannian manifold with Ricci curvature bounded below and Laplace operator $Δ$. The paper develops a functional calculus for the cosine family $\cos(t\sqrt Δ)$ which is associated with waves that travel at unit speed. If $f$ is holomorphic on a Venturi shaped region, and $z^kf(z)$ is bounded for some positive integer $k$, then $f({\sqrt Δ})$ defines a bounded linear operator on $L^p({\cal{M}})$ for some $p>2$. For Jacobi hypergroups with invariant measure $m$ the generalized Fourier transform of $f\in L^1(m)$ gives $\hat f\in H^\infty (Σ_ω)$ for some strip $Σ_ω$.…
