Invariant complementation and projectivity in the Fourier algebra

In this paper, we study the ideals in the Fourier algebra of a locally compact group which are complemented by an invariant projection. In particular we show that when is discrete, every ideal which is complemented by a completely bounded projection must be invariantly complemented. Perhaps surprisingly, this result does not depend of the amenability of the group or the algebra, but instead relies on the operator biprojectivity of the Fourier algebra for a discrete group.

     

On Fourier algebra homomorphisms

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 we show that it can be described, in terms of a piecewise affine map with Y in the coset ring of H, as follows

     

Operator biflatness of the Fourier algebra and approximate indicators for subgroups

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 rôle 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 , the diagonal in G×G fails to have an approximate indicator.     

Biflatness and biprojectivity of the Fourier algebra

We show that the biflatness—in the sense of A. Ya. Helemskiĭ—of the Fourier algebra A(G) of a locally compact group G forces G to either have an abelian subgroup of finite index or to be non-amenable without containing as a closed subgroup. An analogous dichotomy is obtained for biprojectivity. Received: 4 August 2008     

Operator space structure on Feichtinger's Segal algebra

We extend the definition, from the class of abelian groups to a general locally compact group G, of Feichtinger's remarkable Segal algebra S₀(G). In order to obtain functorial properties for non-abelian groups, in particular a tensor product formula, we endow S₀(G) with an operator space structure. With this structure S₀(G) is simultaneously an operator Segal algebra of the Fourier algebra A(G), and of the group algebra L¹(G). 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 S₀(G) is an invariant for G when it is treated simultaneously as a pointwise algebra and a convolutive algebra.     

Operator weak amenability of the Fourier algebra

We show that for any locally compact group , the Fourier algebra is operator weakly amenable.

     

On the connection between sets of operator synthesis and sets of spectral synthesis for locally compact groups

We extend the results by Froelich and Spronk and Turowska on the connection between operator synthesis and spectral synthesis for A(G) to second countable locally compact groups G. This gives us another proof that one-point subset of G is a set of spectral synthesis and that any closed subgroup is a set of local spectral synthesis. Furthermore, we show that “non-triangular” sets are strong operator Ditkin sets and we establish a connection between operator Ditkin sets and Ditkin sets. These results are applied to prove that any closed subgroup of G is a local Ditkin set.     

Module operator virtual diagonals on the Fourier algebra of an inverse semigroup

For an amenable inverse semigroup S with the set of idempotents E and a minimal idempotent, we explicitly construct a contractive and positive module operator virtual diagonal on the Fourier algebra A(S), as a completely contractive Banach algebra and operator module over \(\ell ^1(E)\). This generalizes a well known result of Zhong-Jin Ruan on operator amenability of the Fourier algebra of a (discrete) group Ruan (Am J Math 117:1449–1474, 1995).     

Complemented ideals in the Fourier algebra of a locally compact group

In this paper we provide a necessary condition for a closed ideal in the Fourier algebra of a locally compact amenable group to be completely complemented. The classification of completely complemented ideals is completed in the case of an amenable discrete group. We also investigate the ideals possessing a bounded approximate identity.

     

Weak amenability and the second dual of the Fourier algebra

Let be a locally compact group. We will consider amenability and weak amenability for Banach algebras which are quotients of the second dual of the Fourier algebra. In particular, we will show that if is weakly amenable, then has no infinite abelian subgroup.

     

Weakly compact homomorphisms of regular Banach algebras

Let A be a uniformly regular Ditkin algebra. It is shown that every weakly compact homomorphism of A into a Banach algebra is finite-dimensional.     

Weak spectral synthesis in commutative Banach algebras. II

Let A be a semisimple and regular commutative Banach algebra with structure space Δ(A). 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 Δ(A). 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.     

Kazhdan's Property (T) for the unitary group of aseparable Hilbert space

We show that the unitary group of a separable Hilbert space has Kazhdan's Property (T), when it is equipped with the strong operator topology. More precisely, for every integer m 2, we give an explicit Kazhdan set consisting of m unitary operators and determine an optimal Kazhdan constant for this set. Moreover, we show that a locally compact group with Kazhdan's Property (T) has a finite Kazhdan set if and only if its Bohr compactification has a finite Kazhdan set. As a consequence, if a locally compact group with Property (T) is minimally almost periodic, then it has a finite Kazhdan set.     

Weak spectral synthesis in commutative Banach algebras

Let A be a semisimple and regular commutative Banach algebra with structure space Δ(A). Generalizing the notion of spectral sets in Δ(A), 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 A(G) of a locally compact group G if and only if G is discrete.     

Amenability and weak amenability of the Fourier algebra

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.Research supported by NSERC under grant no. 90749-00.Research supported by NSERC under grant no. 227043-00.     

On {\phi}-contractibility of the Lebesgue–Fourier algebra of a locally compact group

For a locally compact group G, we present some characterizations for f{\phi}-contractibility of the Lebesgue–Fourier algebra LA(G){\mathcal{L}A(G)} endowed with convolution or pointwise product.     

Symmetric nonself-adjoint operators in an enveloping algebra

Nelson and Stinespring proved that in any unitary representation of a Lie group with compact Lie algebra the representation of Hermitian elements in the enveloping algebra are essentially self-adjoint. If the Lie algebra is noncompact, we construct in its enveloping algebra a Hermitian element u such that in any locally faithful unitary representation the representative of u has no self-adjoint extension.     

On the strict,uniform and compact-open topologies on an algebra

We introduce and study strict, uniform, and compact-open locally convex topologies on an algebra \({\mathcal {B}},\) by the fundamental system of seminorms of a locally convex subalgebra \(({\mathcal {A},p}_\alpha )\). Moreover, we investigate when \({\mathcal {B}}\) is a locally convex algebra with respect to these topologies. Furthermore, we generalize an essential result related to derivations, from Banach to the Fréchet case. Finally we provide a useful example in this field.     

Compact right multipliers on a Banach algebra related to locally compact semigroups

For a locally compact semigroup \({\mathcal{S}}\), let \(L_{0}^{\infty}({\mathcal{S}},M_{a}({\mathcal{S}}))\) be the Banach space of all μ-measurable (\(\mu\in M_{a}({\mathcal{S}})\)) functions vanishing at infinity, where \(M_{a}({\mathcal{S}})\) denotes the algebra of all measures in the measure algebra \(M({\mathcal{S}})\) of \({\mathcal{S}}\) with continuous translations. Here, we study right compact multipliers on the Banach algebra \(L_{0}^{\infty}({\mathcal{S}},M_{a}({\mathcal{S}}))^{*}\) equipped with an Arens product.