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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 an open question of Anantharaman--Delaroche on the equivalence of inner amenability and Property (W). In the bimodule setting, we show that relative operator 1-biflatness of $A(G)$ is equivalent to the existence of a contractive approximate indicator for the diagonal $G_\Delta$ in the Fourier--Stieltjes algebra $B(G\times G)$, thereby establishing the converse to a result of Aristov, Runde, and Spronk. We conjecture that relative $1$-biflatness of $A(G)$ is equivalent to the existence of a quasi-central bounded approximate identity in $L^1(G)$, that is, $G$ is QSIN, and verify the conjecture in many special cases. We finish with an application to the operator homology of $A_{cb}(G)$, giving examples of weakly amenable groups for which $A_{cb}(G)$ is not operator amenable., Comment: v3: final version, 22 pages

Let Γ be a locally compact group. We answer two questions left open in [8] and [10]: (i) For abelian Γ, we prove that if χS∈B(Γ) is an idempotent with norm ‖χS‖

Let G be a locally compact group and 1 ≤ p ∞ . A continuous unitary representation π : G → B ( H ) of G is an L p -representation if the matrix coefficient functions s ↦ 〈 π ( s ) x , x 〉 lie in L p ( G ) for sufficiently many x ∈ H . The L p -Fourier algebra A L p ( G ) is defined 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 Fourier–Stieltjes algebra containing the Fourier algebra A ( G ) . In this paper we investigate how these spaces reflect properties of the underlying group and study the structural properties of these algebras. As an application of this theory, we characterize the Fourier–Stieltjes ideals of SL ( 2 , R ) .

This paper is concerned with weak⁎ closed masa-bimodules generated by A ( G ) -invariant subspaces of VN ( G ) . An annihilator formula is established, which is used to characterise the weak⁎ closed subspaces of B ( L 2 ( G ) ) which are invariant under both Schur multipliers and a canonical action of M ( G ) on B ( L 2 ( G ) ) via completely bounded maps. We study the special cases of extremal ideals with a given null set and, for a large class of groups, we establish a link between relative spectral synthesis and relative operator synthesis.

We introduce the class of Beurling–Fourier algebras on locally compact groups and show that they are non-commutative analogs of classical Beurling algebras. We obtain various results with regard to the operator amenability, operator weak amenability and Arens regularity of Beurling–Fourier algebras on compact groups and show that they behave very similarly to the classical Beurling algebras of discrete groups. We then apply our results to study explicitly the Beurling–Fourier algebras on SU ( 2 ) , the 2 × 2 special unitary group. We demonstrate that how Beurling–Fourier algebras are closely connected to the amenability of the Fourier algebra of SU ( 2 ) . Another major consequence of our results is that our investigation allows us to construct families of unital infinite-dimensional closed Arens regular subalgebras of the Fourier algebra of certain products of SU ( 2 ) .

We study power boundedness in the Fourier and Fourier–Stieltjes algebras, A ( G ) and B ( G ) , of a locally compact group G as well as in some other commutative Banach algebras. The main results concern the question of when all elements with spectral radius at most one in any of these algebras are power bounded, the characterization of power bounded elements in A ( G ) and B ( G ) and also the structure of the Gelfand transform of a single power bounded element.

This paper has a two-fold purpose. Let 1 p ∞ . We first introduce the p -operator space injective tensor product and study various properties related to this tensor product, including the p -operator space approximation property, for p -operator spaces on L p -spaces. We then apply these properties to the study of the pseudofunction algebra PF p ( G ) , the pseudomeasure algebra PM p ( G ) , and the Figa–Talamanca–Herz algebra A p ( G ) of a locally compact group G . We show that if G is a discrete group, then most of approximation properties for the reduced group C ∗ -algebra C λ ∗ ( G ) , the group von Neumann algebra VN ( G ) , and the Fourier algebra A ( G ) (related to amenability, weak amenability, and approximation property of G ) have the natural p -analogues for PF p ( G ) , PM p ( G ) , and A p ( G ) , respectively. The p -completely bounded multiplier algebra M cb A p ( G ) plays an important role in this work.

We continue the investigation of notions of approximate amenability that were introduced in work of the second and third authors. It is shown that every boundedly approximately contractible Banach algebra has a bounded approximate identity. Among our other results, it is shown that the Fourier algebra of the free group on two generators is not approximately amenable. Further examples are obtained of ${\ell}^1$-semigroup algebras which are approximately amenable but not amenable; using these, we show that bounded approximate amenability need not imply sequential approximate amenability. Results are also given for Segal subalgebras of $L^1(G)$, where $G$ is a locally compact group, and the algebras $PF_p(\Gamma)$ of $p$-pseudofunctions on a discrete group $\Gamma$ (of which the reduced $C^*$-algebra is a special case)., Comment: 35 pages, revision of Jan '08 preprint. Abstract and MSC added; bibliograpy updated; slight tweaks to Section 4; and correction of a few typos. The final version is to appear in J. Funct. Anal

For a locally compact group G, let XG be one of the following introverted subspaces of VN(G): UCB(Gˆ), the C∗-algebra of uniformly continuous functionals on A(G); W(Gˆ), the space of weakly almost periodic functionals on A(G); or Mρ∗(G), the C∗-algebra generated by the left regular representation on the measure algebra of G. We discuss the extension of homomorphisms of (reduced) Fourier–Stieltjes algebras on G and H to cb-norm preserving, weak∗–weak∗-continuous homomorphisms of XG∗ into XH∗, where (XG,XH) is one of the pairs (UCB(Gˆ),UCB(Hˆ)), (W(Gˆ),W(Hˆ)), or (Mρ∗(G),Mρ∗(H)). When G is amenable, these extensions are characterized in terms of piecewise affine maps.

Keywords: locally compact groups ; amenable groups ; Fourier algebra ; Figa-Talamanca algebra ; convolution operators ; pseudomeasures Reference CAHRU-ARTICLE-2007-001doi:10.1016/j.jfa.2007.03.003View record in Web of Science Record created on 2008-12-10, modified on 2016-08-08

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 first author. We also obtain a description of all the idempotents in the Fourier-Stieltjes algebras which are contractive or positive definite., 19 pages

Let G be a locally compact group and let B(G) be the dual space of C∗(G), the group C∗ algebra of G. The Fourier algebra A(G) is the closed ideal of B(G) generated by elements with compact support. The Fourier algebras have a natural operator space structure as preduals of von Neumann algebras. Given a completely bounded algebra homomorphism φ:A(G)→B(H) we show that it can be described, in terms of a piecewise affine map α:Y→G with Y in the coset ring of H, as followsφ(f)=f∘αonY,0offYwhen G is discrete and amenable. This extends a similar result by Host. We also show that in the same hypothesis the range of a completely bounded algebra homomorphism φ:A(G)→A(H) is as large as it can possibly be and it is equal to a well determined set. The same description of the range is obtained for bounded algebra homomorphisms, this time when G and H are locally compact groups with G abelian.

Column and row operator spaces—which we denote by COL and ROW, respectively—over arbitrary Banach spaces were introduced by the first-named author; for Hilbert spaces, these definitions coincide with the usual ones. Given a locally compact group G and p,p′∈(1,∞) with 1 p + 1 p′ =1 , we use the operator space structure on CB ( COL (L p′ (G))) to equip the Figa-Talamanca–Herz algebra Ap(G) with an operator space structure, turning it into a quantized Banach algebra. Moreover, we show that, for p⩽q⩽2 or 2⩽q⩽p and amenable G, the canonical inclusion Aq(G)⊂Ap(G) is completely bounded (with cb-norm at most K G 2 , where K G is Grothendieck's constant). As an application, we show that G is amenable if and only if Ap(G) is operator amenable for all—and equivalently for one—p∈(1,∞); this extends a theorem by Ruan.

Let G be a locally compact amenable group, B ( G ) its Fourier–Stieltjes algebra and I be a closed ideal of it. In this paper we prove the following result: The ideal I has a unit element iff it is principal. This is the noncommutative version of the Glicksberg–Host–Parreau Theorem. The paper also contains an abstract version of this theorem.

We make use of the operator space structure of the Fourier algebra A(G) of an amenable locally compact group to prove that if H is any closed subgroup of G, then the ideal I(H) consisting of all functions in A(G) vanishing on H has a bounded approximate identity. This result allows us to completely characterize the ideals of A(G) with bounded approximate identities. We also show that for several classes of locally compact groups, including all nilpotent groups, I(H) has an approximate identity with norm bounded by 2, the best possible norm bound.

For a closed subgroup H of a locally compact group G consider the property that the continuous positive definite functions on G which are identically one on H separate points in G \ H from points in H . We prove a structure theorem for almost connected groups having this separation property for every closed subgroup. Also, when a pair ( G , H ) has this separation property, there are interesting consequences in the ideal theory of the Fourier algebra of G .

The classical definitions of the Fourier algebras B ( G ) and A ( G ) of a locally compact group are extended to an arbitrary measured groupoid G . Dualities are established between B ( G ) and A ( G ) and the convolution algebras C * μ ( G ) and VN ( G ) in the framework of operator modules. They are used to generalize results of Varopoulos and Pisier about Littlewood functions and completely bounded multipliers.

In this paper we study the question whether certain non-semisimple, commutative Banach algebras have a Wedderburn decomposition U = B ⊕ rad U , where B is a subalgebra of U . Let A(G) be the Fourier algebra of a locally compact abelian group G, and let E be a closed subset of G. Let J(E) be the smallest closed ideal in A(G) whose hull is E. We prove that, when E is a set of non-synthesis, the quotient algebra A(G) J(E) never has a Wedderburn decomposition even in the purely algebraic sense. This result is extended to cover certain Beurling algebras Ax(Rk) and Ax(Tk).

We make precise the following statements: B(G), the Fourier-Stieltjes algebra of locally compact group G, is a dual of G and vice versa. Similarly, A(G), the Fourier algebra of G, is a dual of G and vice versa. We define an abstract Fourier (respectively, Fourier-Stieltjes) algebra; we define the dual group of such a Fourier (respectively, Fourier-Stieltjes) algebra; and we prove the analog of the Pontriagin duality theorem in this context. The key idea in the proof is the characterization of translations of B(G) as precisely those isometric automorphisms Φ of B(G) which satisfy ∥ p − eiθΦp ∥2 + ∥ p + eiθΦp ∥2 = 4 for all θ ∈ R and all pure positive definite functions p with norm one. One particularly interesting technical result appears, namely, given x1, x2 ϵ G, neither of which is the identity e of G, then there exists a continuous, irreducible unitary representation π of G (which may be chosen from the reduced dual of G) such that π(x1) ≠ π(e) and π(x2) ≠ π(e). We also note that the group of isometric automorphisms of B(G) (or A(G)) contains as a (“large”) .closed, normal subgroup the topological version of Burnside's “holomorph of G.”

In this paper we generalize the classical Bernstein theorem concerning the absolute convergence of the Fourier series of Lipschitz functions. More precisely, we consider a group G which is finite dimensional, compact, and separable and has an infinite, closed, totally disconnected, normal subgroup D, such that GD is a Lie group. Using this structure, we define in a natural way the notion of Lipschitz condition, and then prove that a function which satisfies a Lipschitz condition of order greater than (dim G + 1)2 belongs to the Fourier algebra of G.

In this paper we study in the context of compact totally disconnected groups the relationship between the smoothness of a function and its membership in the Fourier algebra GG. Specifically, we define a notion of smoothness which is natural for totally disconnected groups. This in turn leads to the notions of Lipshitz condition and bounded variation. We then give a condition on α which if satisfied implies Lipα(G) ⊂ R(G). On certain groups this condition becomes: α > 12 (Bernstein's theorem). We then give a similar condition on α which if satisfied implies that Lipα(G) ∈ BV(G) ⊂ R(G). On certain groups this condition becomes: α > 0 (Zygmund's theorem). Moreover we show that α > 12 is best possible by showing that Lip12(G) ⊄ R(G).

The Fourier-Stieltjes algebra of an arbitrary locally compact group G is the set of all finite, complex-linear combinations of continuous, positive definite functions on G, where addition and multiplication are defined pointwise and a Banach algebra norm (unique up to equivalence) can be specified. Thus, B(G) is a commutative, semisimple Banach algebra with unit. The main result is that B(G1) and B(G2) are isometrically isomorphic as Banach algebras if and only if G1 and G2 are topologically isomorphic as groups. The spectrum of B(G) is characterized as a ∗-semigroup of operators on Hilbert space, and its subgroup of invertible elements (being precisely those unitary elements which “preserve tensor products”) is topologically isomorphic to G. The Fourier algebra A(G) is also shown to characterize G. (A(G) can be defined as the closure in B(G) of the functions in B(G) with compact support.) The representation theory of the lattice of subgroups of G is also studied. The main techniques of investigation come from the theory of C ∗ and W∗-algebras.

We study certain commutative regular semisimple Banach algebras which we call hyper-Tauberian algebras. We first show that they form a subclass of weakly amenable Tauberian algebras. Then we investigate the basic and hereditary properties of them. Moreover, we show that if A is a hyper-Tauberian algebra, then the linear space of bounded derivations from A into any Banach A -bimodule is reflexive. We apply these results to the Figa–Talamanca–Herz algebra A p ( G ) of a locally compact group G for p ∈ ( 1 , ∞ ) . We show that A p ( G ) is hyper-Tauberian if the principal component of G is abelian. Finally, by considering the quantization of these results, we show that for any locally compact group G , A p ( G ) , equipped with an appropriate operator space structure, is a quantized hyper-Tauberian algebra. This, in particular, implies that A p ( G ) is operator weakly amenable.

We investigate if, for a locally compact group $G$, the Fourier algebra $A(G)$ is biflat in the sense of quantized Banach homology. A central role in our investigation is played by the notion of an approximate indicator of a closed subgroup of $G$: The Fourier algebra is operator biflat whenever the diagonal in $G \times G$ has an approximate indicator. Although we have been unable to settle the question of whether $A(G)$ is always operator biflat, we show that, for $G = SL(3,C)$, the diagonal in $G \times G$ fails to have an approximate indicator., 23 pages; more typos removed; references updated

Let G be a compact nonmetrizable topological group whose local weight b(G) has uncountable cofinality. Let H be an amenable locally compact group, A(G×H) the Fourier algebra of G×H, and UC2(G×H) the space of uniformly continuous functionals in VN(G×H)=A(G×H)∗. We use weak factorization of operators in the group von Neumann algebra VN(G×H) to prove that there exist at least 22b(G) left ideals of dimensions at least 22b(G) in A(G×H)∗∗ and in UC2(G×H)∗. We show that every nontrivial right ideal in A(G×H)∗∗ and in UC2(G×H)∗ has dimension at least 22b(G).

Let F1(Rn) denote the Fourier algebra on Rn, and D(Rn) the space of test functions on Rn. A closed subset E of Rn is said to be of spectral synthesis if the only closed ideal J in F1(Rn) which has E as its hull h(J)={x ϵ Rn:f(x)=0 for all f ϵ J} is the ideal k(E)={fϵF1(Rn):f(E)=0}. We consider sufficiently regular compact subsets of smooth submanifolds of Rn with constant relative nullity. For such sets E we give an estimate of the degree of nilpotency of the algebra (k(E)∩D(Rn))−j(E), where j(E) denotes the smallest closed ideal in F1(Rn) with hull E. Especially in the case of hypersurfaces this estimate turns out to be exact. Moreover for this case we prove that k(E)∩D(Rn) is dense in k(E). Together this solves the synthesis problem for such sets.