The Fourier-Stieltjes and Fourier algebras B(G), A(G) for a general locally compact group G, first studied by P. Eymard, have played an important role in harmonic analysis and in the study of operator algebras generated by G. Recently, there has been interest in developing versions of these algebras for locally compact groupoids, justification being that, just as in the group case, the algebras should play a useful role in the study of groupoid operator algebras. Versions of these algebras for the locally compact groupoid case appear in three related theories: (1) a measured groupoid theory (J. Renault), (2) a Borel theory (A. Ramsay and M. Walter), and (3) a continuous theory (A. Paterson). The present paper is expository in character. For motivational reasons, it starts with a description of the theory of B(G), A(G) in the locally compact group case, before discussing these three realted theories. Some open questions are also raised.
This paper gives a first step toward extending the theory of Fourier-Stieltjes algebras from groups to groupoids. If G is a locally compact (second countable) groupoid, we show that B(G), the linear span of the Borel positive definite functions on G, is a Banach algebra when represented as an algebra of completely bounded maps on a C^*-algebra associated with G. This necessarily involves identifying equivalent elements of B(G). An example shows that the linear span of the continuous positive definite functions need not be complete. For groups, B(G) is isometric to the Banach space dual of C^*(…
It is a well-known result of Eymard that the Fourier-Stieltjes algebra of a locally compact group $G$ can be identified with the dual of the group $\cs$ $C^{*}(G)$. A corresponding result for a locally compact groupoid $G$ has been investigated by Renault, Ramsay and Walter. We show that the Fourier-Stieltjes algebra $B_μ(G)$ of $G$ (with respect to a quasi-invariant measure $μ$ on the unit space $X$ of $G$) can be characterized in operator space terms as the dual of the Haagerup tensor product $\ov{L^{2}(X,μ)}^{r}\otimes_{hA}C^{*}(G,μ)\otimes_{h A}L^2(X,μ)^c$ and as the space of completely bo…
Let $G$ be a locally compact group and $1\leq p<\infty$. A continuous unitary representation $π\!: G\to B(\mathcal{H})$ of $G$ is an $L^p$-representation if the matrix coefficient functions $s\mapsto \langle π(s)x,x\rangle$ lie in $L^p(G)$ for sufficiently many $x\in \mathcal{H}$. Brannan and Ruan defined the $L^p$-Fourier algebra $A_{L^p}(G)$ to be the set of matrix coefficient functions of $L^p$-representations. Similarly, the $L^p$-Fourier-Stieltjes algebra $B_{L^p}(G)$ is defined to be the weak*-closure of $A_{L^p}(G)$ in the Fourier-Stieltjes algebra $B(G)$. These are always ideals in the…
In analogy with the Fourier-Stieltjes algebra of a group, we associate to a unital discrete twisted C*-dynamical system a Banach algebra whose elements are coefficients of equivariants representations of the system. Building upon our previous work, we show that this Fourier-Stieltjes algebra embeds continuously in the Banach algebra of completely bounded multipliers of the (reduced or full) C*-crossed product of the system. We also introduce a notion of positive definiteness and prove a Gelfand-Raikov type theorem allowing us to describe the Fourier-Stieltjes algebra of a system in a more intr…
In this paper we start studying spectral properties of the Fourier-Stieltjes algebras, largely following Zafran's work on the algebra of measures on a locally compact group. We show that for a large class of discrete groups the Wiener-Pitt phenomenon occurs, i.e. the spectrum of an element of the Fourier-Stieltjes algebra is not captured by its range. We also investigate the notions of absolute continuity and mutual singularity in this setting; non-commutativity forces upon us two distinct versions of support of an element, indicating a crucial difference between this setup and the realm of Ab…
We consider the Fourier-Stietljes algebra B(G) of a locally compact group G. We show that operator amenablility of B(G) implies that a certain semitolpological compactification of G admits only finitely many idempotents. In the case that G is connected, we show that operator amenability of B(G) entails that $G$ is compact.
We continue our study of the Fourier-Stieltjes algebra associated to a twisted (unital, discrete) C*-dynamical system and discuss how the various notions of equivalence of such systems are reflected at the algebra-level. As an application, we show that the amenability of a system, as defined in our previous work, is preserved under Morita equivalence.
We give an example of a non-compact, locally compact group $G$ such that its Fourier-Stieltjes algebra $B(G)$ is operator amenable. Furthermore, we characterize those $G$ for which $A^*(G)$ - the spine of $B(G)$ as introduced by M. Ilie and the second named author - is operator amenable and show that $A^*(G)$ is operator weakly amenable for each $G$.
We study the closed algebra B_I(G) generated by the idempotents in the Fourier-Stieltjes algebra of a locally compact group G. We show that it is a regular Banach algebra with computable spectrum G^I, which we call the idempotent compactification of G. For any locally compact groups G and H, we show that B_I(G) is completely isometrically isomorphic to B_I(H) exactly when G/G_e= H/H_e, where G_e and H_e are the connected components of the identities. We compute some examples to illustrate out results.
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…
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 define the spine A*(G) of the Fourier-Stieltjes algebra B(G) of a locally compact group G. A*(G) is graded over a certain semi-lattice, that of non-quotient locally precompact topologies on G. We compute the spine's spectrum G*, which admits a semi-group structure. We discuss homomorphisms from A*(G) to B(H) where H is another locally compact group; and we show that A*(G) contains the image of every completely bounded homomorphism from the Fourier algebra A(H) of any amenable group H. We also show that A*(G) contains all of the idempotents in B(G). Finally, we compute examples for vector gr…
The notion of operator amenability was introduced by Z.-J. Ruan in 1995. He showed that a locally compact group G is amenable if and only if its Fourier algebra A(G) is operator amenable. In this paper, we investigate the operator amenability of the Fourier-Stieltjes algebra B(G) and of the reduced Fourier-Stieltjes algebra B_r(G). The natural conjecture is that any of these algebras is operator amenable if and only if G is compact. We partially prove this conjecture with mere operator amenability replaced by operator C-amenability for some constant C < 5. In the process, we obtain a new decom…
We introduce and investigate using Hilbert modules the properties of the Fourier algebra A(G) for a locally compact groupoid G. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This includes as a special case the classical duality theorem for locally compact groups proved by P. Eymard.
We introduce and investigate using Hilbert modules the properties of the {\em Fourier algebra} $A(G)$ for a locally compact groupoid $G$. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This includes as a special case the classical duality theorem for locally compact groups proved by P. Eymard.
In a recent paper on exotic crossed products, we included a lemma concerning ideals of the Fourier-Stieltjes algebra. Buss, Echterhoff, and Willett have pointed out to us that our proof of this lemma contains an error. In fact, it remains an open question whether the lemma is true as stated. In this note we indicate how to contain the resulting damage. Our investigation of the above question leads us to define two properties \emph{ordered} and \emph{weakly ordered} for invariant ideals of Fourier-Stieltjes algebras, and we initiate a study of these properties.
We determine the structure of the weak*-closed $G$-invariant ideals in the Fourier-Stieltjes algebra $B(G)$ of certain groups $G$ by means of a $K$-theoretical obstruction. The groups to which this applies are groups whose only irreducible unitary representations that are not weakly contained in the left regular representation are class-one representations. In particular, this is the case for the groups $\mathrm{SL}(2,\mathbb{R})$ and $\mathrm{SL}(2,\mathbb{C})$, which we consider as explicit examples.
For a Fell bundle $\mathcal{B}=\left\{B_{s}\right\}_{s \in G}$ over a discrete group $G$, we use representations theory of $\mathcal{B}$ to construct the Fourier and Fourier-Stieltjes spaces $A(\mathcal{B})$ and $B(\mathcal{B})$ of $\mathcal B$. When $\mathcal B$ is saturated we show $B(\mathcal{B})$ is canonically isomorphic to the dual space of the cross sectional $C^{*}$-algebra $C^{*}(\mathcal{B})$ of $\mathcal{B}$. When there is a compatible family of co-multiplications on the fibers we show that $B(\mathcal{B})$ and $A(\mathcal{B})$ are Banach algebras. This holds in particular if either…
In a series of papers, we have shown that from the representation theory of a compact groupoid one can reconstruct the groupoid using the procedure similar to the Tannaka-Krein duality for compact groups. In this part we study the Fourier and Fourier-Plancherel transforms and prove the Plancherel theorem for compact groupoids. We also study the central functions in the algebra of square integrable functions on the isotropy groups.
For locally compact groups G and H let A(G) denote the Fourier algebra of G and B(H) the Fourier-Stieltjes algebra of H. Any continuous piecewise affine map alpha:Y -> G (where Y is an element of the open coset ring of H) induces a completely bounded homomorphism Phi_alpha:A(G) -> B(H) by setting Phi_alpha u(.)=u(alpha(.)) on Y and Phi_alpha u=0 off of Y. We show that if G is amenable then any completely bounded homomorphism Phi:A(G) -> B(H) is of this form; and this theorem fails if G contains a discrete nonabelian free group. Our result generalises results of P.J. Cohen, B. Host and of the f…
We introduce a class of probability measures whose densities near infinity are mixtures of Pareto distributions. This class can be characterized by the Fourier transform which has a power series expansion including real powers, not only integer powers. This class includes stable distributions in probability and also non-commutative probability theories. We also characterize the class in terms of the Cauchy-Stieltjes transform and the Voiculescu transform. If the stability index is greater than one, stable distributions in probability theory do not belong to that class, while they do in non-com…
As announced in [12], we develop a calculus of Fourier integral G-operators on any Lie groupoid G. For that purpose, we study convolability and invertibility of Lagrangian conic submanifolds of the symplectic groupoid T * G. We also identify those Lagrangian which correspond to equivariant families parametrized by the unit space G (0) of homogeneous canonical relations in (T * Gx \ 0) x (T * G x \ 0). This allows us to select a subclass of Lagrangian distributions on any Lie groupoid G that deserve the name of Fourier integral G-operators (G-FIO). By construction, the class of G-FIO contains t…
The Fourier and Fourier-Stieltjes algebras $A(G)$ and $B(G)$ of a locally compact group $G$ are introduced and studied in 60's by Piere Eymard in his PhD thesis. If $G$ is a locally compact abelian group, then $A(G)\simeq L^1(\hat{G})$, and $B(G)\simeq M(\hat{G})$, via the Fourier and Fourier-Stieltjes transforms, where $\hat{G}$ is the Pontryagin dual of $G$. Recently these algebras are defined on a (topological or measured) groupoid and have shown to share many common features with the group case. This is the last in a series of papers in which we have investigated a "restricted" form of the…
Let $G$ be a locally compact group. We show that its Fourier algebra $A(G)$ is amenable if and only if $G$ has an abelian subgroup of finite index, and that its Fourier-Stieltjes algebra $B(G)$ is amenable if and only if $G$ has a compact, abelian subgroup of finite index. We then show that $A(G)$ is weakly amenable if the component of the identity of $G$ is abelian, and we prove some partial results towards the converse.
We introduce the notion of the Fourier and Fouier-Stieltjes algebra of a topological *-semigroup and show that these are commutative Banach algebras. For a class of foundation semigroups, we show that these are preduals of von Neumann algebras.
The classical Hausdorff-Young inequality for locally compact abelian groups states that, for $1\le p\le 2$, the $L^p$-norm of a function dominates the $L^q$-norm of its Fourier transform, where $1/p+1/q=1$. By using the theory of non-commutative $L^p$-spaces and by reinterpreting the Fourier transform, R. Kunze (1958) [resp. M. Terp (1980)] extended this inequality to unimodular [resp. non-unimodular] groups. The analysis of the $L^p$-spaces of the von Neumann algebra of a measured groupoid provides a further extension of the Hausdorff-Young inequality to measured groupoids.
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 $…
Using the calculus of Fourier integral operators on Lie groupoids developped in [18], we study the fundamental solution of the evolution equation ($\partial$ $\partial$t + iP)u = 0 where P is a self adjoint elliptic order one G-pseudodifferential operator on the Lie groupoid G. Along the way, we continue the study of distributions on Lie groupoids done in [17] by adding the reduced C *-algebra of G in the picture and we investigate the local nature of the regularizing operators of [32].
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 Haa…
We introduce C*-pseudo-multiplicative unitaries and concrete Hopf C*-bimodules for the study of quantum groupoids in the setting of C*-algebras. These unitaries and Hopf C*-bimodules generalize multiplicative unitaries and Hopf C*-algebras and are analogues of the pseudo-multiplicative unitaries and Hopf--von Neumann-bimod-ules studied by Enock, Lesieur and Vallin. To each C*-pseudo-multiplicative unitary, we associate two Fourier algebras with a duality pairing, a C*-tensor category of representations, and in the regular case two reduced and two universal Hopf C*-bimodules. The theory is illu…
We study various operator homological properties of the Fourier algebra $A(G)$ of a locally compact group $G$. Establishing the converse of two results of Ruan and Xu, we show that $A(G)$ is relatively operator 1-projective if and only if $G$ is IN, and that $A(G)$ is relatively operator 1-flat if and only if $G$ is inner amenable. We also exhibit the first known class of groups for which $A(G)$ is not relatively operator $C$-flat for any $C\geq1$. As applications of our techniques, we establish a hereditary property of inner amenability, answer an open question of Lau and Paterson, and answer…
We review the properties of transversality of distributions with respect to submersions. This allows us to construct a convolution product for a large class of distributions on Lie groupoids. We get a unital involutive algebra $\cE\_{r,s}'(G,Ω^{1/2})$ enlarging the convolution algebra $C^\infty\_c(G,Ω^{1/2})$ associated with any Lie groupoid $G$. We prove that $G$-operators are convolution operators by transversal distributions. We also investigate the microlocal aspects of the convolution product. We give conditions on wave front sets sufficient to compute the convolution product and we show …
For a locally compact Hausdorff semigroup S, the $L^\infty$ representation algebra R(S) was extensively studied by Dunkl and Ramirez. The Fourier-Stieltjes algebra F(S) of a topological semigroup was studied by Lau. The aim of this paper is to investigate these two algebras and study the amenability of them with respect to the structure of S.
Completely positive and completely bounded mutlipliers on rigid $C^{\ast}$-tensor categories were introduced by Popa and Vaes. Using these notions, we define and study the Fourier-Stieltjes algebra, the Fourier algebra and the algebra of completely bounded multipliers of a rigid $C^{\ast}$-tensor category. The rich structure that these algebras have in the setting of locally compact groups is still present in the setting of rigid $C^{\ast}$-tensor categories. We also prove that Leptin's characterization of amenability still holds in this setting, and we collect some natural observations on pro…
This is a short survey on idempotent states on locally compact groups and locally compact quantum groups. The central topic is the relationship between idempotent states, subgroups and invariant C*-subalgebras. We concentrate on recent results on locally compact quantum groups, but begin with the classical notion of idempotent probability measure. We also consider the `intermediate' case of idempotent states in the Fourier--Stieltjes algebra: this is the dual case of idempotent probability measures and so an instance of idempotent states on a locally compact quantum group.
Operator systems and positive extensions over discrete groups
National and Kapodistrian University of Athens（雅典国立卡波迪斯特里安大学）；Max Planck Institute for Mathematics（马克斯·普朗克数学研究所）；University of Delaware（特拉华大学）；Radboud University Nijmegen（奈梅亨拉德堡德大学）
A QSIN group is a locally compact group $G$ whose group algebra $L^1(G)$ admits a quasi-central bounded approximate identity. Examples of QSIN groups include every amenable group and every discrete group. It is shown that if $G$ is a QSIN group, $H$ is a closed subgroup of $G$, and $π$ is a unitary representation of $H$, then $π$ is weakly contained in $(\mathrm{Ind}_H^Gπ)|_H$. This provides a powerful tool in studying the C*-algebras of QSIN groups. In particular, it is shown that if $G$ is a QSIN group which contains a copy of $\mathbb F_2$ as a closed subgroup, then $C^*(G)$ is not locally …
We prove that if $ρ$ is an adapted positive definite function in the Fourier--Stieltjes algebra $B(G)$ of a locally compact group $G$ with $\|ρ\|_{B(G)}=1$, then the iterated powers $(ρ^n)$ converge to zero in the weak* topology $σ(B(G) , C^*(G))$. Moreover, if $ρ$ is irreducible, we prove that $(ρ^n)$ as a sequence of u.c.p. maps on the group $C^*$-algebra converges to zero in the strong operator topology.
In 1972, B. E. Johnson proved that a locally compact group $G$ is amenable if and only if certain Hochschild cohomology groups of its convolution algebra $L^1(G)$ vanish. Similarly, $G$ is compact if and only if $L^1(G)$ is biprojective: In each case, a classical property of $G$ corresponds to a cohomological propety of $L^1(G)$. Starting with the work of Z.-J. Ruan in 1995, it has become apparent that in the non-commutative setting, i.e. when dealing with the Fourier algebra $A(G)$ or the Fourier-Stieltjes algebra $B(G)$, the canonical operator space structure of the algebras under considerat…
We study a generalisation of operator spaces modelled on $L_p$ spaces, instead of Hilbert spaces, using the notion of $p$-complete boundedness, as studied by Pisier and Le Merdy. We show that the Figá-Talamanca-Herz Algebras $A_p(G)$ becomes quantised Banach algebras in this framework, and that the cohomological notion of amenability of these algebras corresponds to amenability of the locally compact group $G$. We thus argue that we have presented a generalised of the use of operator spaces in studying the Fourier algebra $A(G)$, in the spirit of Ruan. Finally, we show that various notions of …
We show that for a locally compact group G there is a one-to-one correspondence between G-invariant weak*-closed subspaces E of the Fourier-Stieltjes algebra B(G) containing B_r(G) and quotients C*_E(G) of C*(G) which are intermediate between C*(G) and the reduced group algebra C*_r(G). We show that the canonical comultiplication on C*(G) descends to a coaction or a comultiplication on C*_E(G) if and only if E is an ideal or subalgebra, respectively. When αis an action of G on a C*-algebra B, we define "E-crossed products" B\rtimes_{α,E} G lying between the full crossed product and the reduced…
With every locally compact group $G$, one can associate several interesting bi-invariant subspaces $X(G)$ of the weakly almost periodic functions $\mathrm{WAP}(G)$ on $G$, each of which captures parts of the representation theory of $G$. Under certain natural assumptions, such a space $X(G)$ carries a unique invariant mean and has a natural predual, and we view the weak$^*$-continuity of this mean as a rigidity property of $G$. Important examples of such spaces $X(G)$, which we study explicitly, are the algebra $M_{\mathrm{cb}}A_p(G)$ of $p$-completely bounded multipliers of the Figà-Talamanca…
If a locally compact group G acts on a C*-algebra B, we have both full and reduced crossed products, and each has a coaction of G. We investigate "exotic" coactions in between, that are determined by certain ideals E of the Fourier-Stieltjes algebra B(G) -- an approach that is inspired by recent work of Brown and Guentner on new C*-group algebra completions. We actually carry out the bulk of our investigation in the general context of coactions on a C*-algebra A. Buss and Echterhoff have shown that not every coaction comes from one of these ideals, but nevertheless the ideals do generate a wid…
We study the group $C^*$-algebras $C^*_{L^{p+}}(G)$ - constructed from $L^p$-integrability properties of matrix coefficients of unitary representations - of locally compact groups $G$ acting on (semi-)homogeneous trees of sufficiently large degree. These group $C^*$-algebras lie between the universal and the reduced group $C^*$-algebra. By directly investigating these $L^p$-integrability properties, we first show that for every non-compact, closed subgroup $G$ of the automorphism group $\mathrm{Aut}(T)$ of a (semi-)homogeneous tree $T$ that acts transitively on the boundary $\partial T$ and ev…
We generalise the definition of a group algebra so that it makes sense for non-locally compact topological groups, in particular, we require that the representation theory of the group algebra is isomorphic (in the sense of Gelfand-Raikov) to the continuous representation theory of the group, or to some other important subset of representations. We prove that a group algebra if it exists, is always unique up to isomorphism. From examples, group algebras do not always exist for non-locally compact groups, but they do exist for some. We define a convolution on the dual of the Fourier-Stieltjes a…
A one to one correspondence between shifts of group-like projections on a locally compact quantum group ${\mathbb{G}}$ which are preserved by the scaling group and contractive idempotent functionals on the dual $\hat{\mathbb{G}}$ is established. This is a generalization of the Illie-Spronk's correspondence between contractive idempotents in the Fourier-Stieltjes algebra of a locally compact group $G$ and cosets of open subgroups of $G$. We also establish a one to one correspondence between non-degenerate, integrable, ${\mathbb{G}}$-invariant ternary rings of operators $X\subset L^\infty({\math…
