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We address two errors made in our paper [7]. The most significant error is in Theorem 1.1. We repair this error, and show that the main result, Theorem 2.5 of [7] , is true. The second error is in one of our examples, Remark 2.4 (iv), and we partially resolve it. [ABSTRACT FROM AUTHOR]

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 [35] , 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 ≥ 1 . As applications of our techniques, we establish a hereditary property of inner amenability, answer an open question of Lau and Paterson [24] , and answer an open question of Anantharaman-Delaroche [1] 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 Δ in the Fourier–Stieltjes algebra B ( G × G ) , thereby establishing the converse to a result of Aristov, Runde, and Spronk [3] . 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 c b ( G ) , giving examples of weakly amenable groups for which A c b ( G ) is not operator amenable. [ABSTRACT FROM AUTHOR]

We show that for a locally compact group G , amongst a class which contains amenable and small invariant neighbourhood groups, 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 ) → B ( H ) admits an invertible S in B ( H ) for which ‖ S ‖ ‖ S − 1 ‖ ≤ ‖ π ‖ cb 2 and S − 1 π ( ⋅ ) S extends to a ⁎-representation of the C*-algebra C 0 ( G ) . This significantly improves some results due to Brannan and Samei (2010) [5] and Brannan, Daws and Samei (2013) [4] . We also note that A ( G ) has completely bounded similarity degree 1 if and only if it is completely isomorphic to an operator algebra if and only if G is finite. [ABSTRACT FROM AUTHOR]

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 χ S ∈ B ( Γ ) is an idempotent with norm ‖ χ S ‖ < 4 3 , then S is the union of two cosets of an open subgroup of Γ. (ii) For general Γ, we prove that if χ S ∈ M c b A ( Γ ) is an idempotent with norm ‖ χ S ‖ c b < 1 + 2 2 , then S is an open coset in Γ. [ABSTRACT FROM AUTHOR]

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 ) . [ABSTRACT FROM AUTHOR]

Abstract: Using techniques of non-abelian harmonic analysis, we construct an explicit, non-zero cyclic derivation on the Fourier algebra of the real group. In particular this provides the first proof that this algebra is not weakly amenable. Using the structure theory of Lie groups, we deduce that the Fourier algebras of connected, semisimple Lie groups also support non-zero, cyclic derivations and are likewise not weakly amenable. Our results complement earlier work of Johnson (1994) [15], Plymen (2001) [18] and Forrest, Samei, and Spronk (2009) [9]. As an additional illustration of our techniques, we construct an explicit, non-zero cyclic derivation on the Fourier algebra of the reduced Heisenberg group, providing the first example of a connected nilpotent group whose Fourier algebra is not weakly amenable. [Copyright &y& Elsevier]

Abstract: This paper is concerned with weak⁎ closed masa-bimodules generated by -invariant subspaces of . An annihilator formula is established, which is used to characterise the weak⁎ closed subspaces of which are invariant under both Schur multipliers and a canonical action of on 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. [Copyright &y& Elsevier]

Abstract: We study power boundedness in the Fourier and Fourier–Stieltjes algebras, and , 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 and and also the structure of the Gelfand transform of a single power bounded element. [Copyright &y& Elsevier]

Abstract: Let A be a semisimple and regular commutative Banach algebra with structure space . Continuing our investigation in [E. Kaniuth, Weak spectral synthesis in commutative Banach algebras, J. Funct. Anal. 254 (2008) 987–1002], we establish various results on intersections and unions of weak spectral sets and weak Ditkin sets in . As an important example, the algebra of n-times continuously differentiable functions is studied in detail. In addition, we prove a theorem on spectral synthesis for projective tensor products of commutative Banach algebras which applies to Fourier algebras of locally compact groups. [Copyright &y& Elsevier]

Abstract: In this paper we investigate when various Banach algebras associated to a locally compact group G have the weak or weak∗ fixed point property for left reversible semigroups. We proved, for example, that if G is a separable locally compact group with a compact neighborhood of the identity invariant under inner automorphisms, then the Fourier–Stieltjes algebra of G has the weak∗ fixed point property for left reversible semigroups if and only if G is compact. This generalizes a classical result of T.C. Lim for the case when G is the circle group T. [Copyright &y& Elsevier]

Abstract: We continue the investigation of notions of approximate amenability that were introduced in work of the second and third authors together with R.J. Loy. It is shown that every boundedly approximately contractible Banach algebra has a bounded approximate identity, and that the Fourier algebra of the free group on two generators is not operator approximately amenable. Further examples are obtained of -semigroup algebras which are approximately amenable but not amenable; using these, we show that bounded approximate contractibility need not imply sequential approximate amenability. Results are also given for Segal algebras on locally compact groups, and algebras of p-pseudo-functions on discrete groups. [Copyright &y& Elsevier]

Abstract: For a locally compact group G, let be one of the following introverted subspaces of : , the -algebra of uniformly continuous functionals on ; , the space of weakly almost periodic functionals on ; or , the -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 into , where is one of the pairs , , or . When G is amenable, these extensions are characterized in terms of piecewise affine maps. [Copyright &y& Elsevier]

Abstract: In this paper, we explore the properties of projections of norm one on general Banach algebras, in particular the relation with conditional expectations for algebras which arise in harmonic analysis. [Copyright &y& Elsevier]

Abstract: Let A be a semisimple and regular commutative Banach algebra with structure space . Generalizing the notion of spectral sets in , the considerably larger class of weak spectral sets was introduced and studied in [C.R. Warner, Weak spectral synthesis, Proc. Amer. Math. Soc. 99 (1987) 244–248]. We prove injection theorems for weak spectral sets and weak Ditkin sets and a Ditkin–Shilov type theorem, which applies to projective tensor products. In addition, we show that weak spectral synthesis holds for the Fourier algebra of a locally compact group G if and only if G is discrete. [Copyright &y& Elsevier]

Abstract: We extend the definition, from the class of abelian groups to a general locally compact group G, of Feichtinger''s remarkable Segal algebra . In order to obtain functorial properties for non-abelian groups, in particular a tensor product formula, we endow with an operator space structure. With this structure is simultaneously an operator Segal algebra of the Fourier algebra , and of the group algebra . We show that this operator space structure is consistent with the major functorial properties: (i) completely isomorphically (operator projective tensor product), if H is another locally compact group; (ii) the restriction map is completely surjective, if H is a closed subgroup; and (iii) is completely surjective, where N is a normal subgroup and . We also show that is an invariant for G when it is treated simultaneously as a pointwise algebra and a convolutive algebra. [Copyright &y& Elsevier]

Abstract: We obtain for H, a closed amenable subgroup of a locally compact group G, the existence of an invariant conditional expectation of onto . As a consequence we prove that H is locally p-Ditkin in G. We also establish relations between , and where is the norm closure, in the set of all bounded operators of , of the set all convolution operators with compact support. [Copyright &y& Elsevier]

Abstract: 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 Figà–Talamanca–Herz algebra of a locally compact group G for . We show that 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, , equipped with an appropriate operator space structure, is a quantized hyper-Tauberian algebra. This, in particular, implies that is operator weakly amenable. [Copyright &y& Elsevier]

Abstract: For locally compact groups G and H let denote the Fourier algebra of G and the Fourier–Stieltjes algebra of H. Any continuous piecewise affine map (where Y is an element of the open coset ring) induces a completely bounded homomorphism by setting on Y and off of Y. We show that if G is amenable then any completely bounded homomorphism is of this form; and this theorem fails if G contains a discrete nonabelian free group. Our result generalises results of Cohen (Amer. J. Math. 82 (1960) 213–226), Host (Bull. Soc. Math. France (1986) 114) and of the first author (J. Funct. Anal. (2004) 213). We also obtain a description of all the idempotents in the Fourier–Stieltjes algebras which are contractive or positive definite. [Copyright &y& Elsevier]

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 followswhen 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. [Copyright &y& Elsevier]

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/p′=1, we use the operator space structure on CB(COL(Lp′(G))) to equip the Figa&grave;-Talamanca–Herz algebra Ap(G) with an operator space structure, turning it into a quantized Banach algebra. Moreover, we show that, for p&les;q&les;2 or 2&les;q&les;p and amenable G, the canonical inclusion Aq(G)⊂Ap(G) is completely bounded (with cb-norm at most KG2, where KG 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. [Copyright &y& Elsevier]

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 roˆle 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×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×G fails to have an approximate indicator. [Copyright &y& Elsevier]

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. [Copyright &y& Elsevier]