Nico Spronk

Measurable Schur Multipliers and Completely Bounded Multipliers of the Fourier Algebras

Let G be a locally compact group, and AðGÞ the Fourier algebra of G from [17]. In [24], Herz de-ned the space of Herz --Schur multipliers, B2ðGÞ. He claimed that this space coincided with the space he denoted VðGÞ in his article [23], but o2ered no proof of this fact. The completely bounded multipliers of the Fourier algebra, McbAðGÞ, were de-ned by De Canniere and Haagerup in [8]. In was shown by Bo_ zejko and Fendler [7] that McbAðGÞ 1⁄4 B2ðGÞ, isometrically. Unfortunately, this result relies on unpublished work of Gilbert (entitled ‘Convolution operators and Banach space tensor products I, II and III’ but not widely available). However, there is a proof of the fact that McbAðGÞ 1⁄4 B2ðGÞ, due to Jolissaint [27], which relies on the representation theorem for completely bounded maps applied to the reduced C -algebra, C rðGÞ. In this paper we prove an analogue of the fact that McbAðGÞ 1⁄4 B2ðGÞ. In fact, our result obtains that fact in the case that G is a discrete group, and improves upon it in the respect that we obtain a complete isometry, where both McbAðGÞ and B2ðGÞ are given natural operator space structures. Moreover, we show that the natural module action of McbAðGÞ on the von Neumann algebra generated by the left regular representation, VNðGÞ, extends completely isometrically to a normal action of McbAðGÞ on all of BðLðGÞÞ, the space of bounded operators on LðGÞ. In fact, the image of McbAðGÞ in the (completely) bounded normal maps on BðLðGÞÞ consists of L1ðGÞ-bimodule maps, and can be characterised as a certain intrinsic ‘invariant part’ amongst those maps. We feel that our approach is interesting in and of itself since it makes systematic use of modern operator space theoretic techniques, pioneered by Blecher, E2ros, Haagerup, Paulsen, Ruan and Smith, amongst others. It thus gives a starting point for obtaining representations of completely bounded multipliers of Kac algebras, after [29], or of multipliers of locally compact quantum groups after Vaes et al. (see [47]). To support our approach, in x 3 we describe the space of measurable Schur multipliers, V1ðX; Þ, for any suitable measure space ðX; Þ. We develop the theory of V1ðX; Þ fully and systematically. Our main tools for doing this are the results of Smith [41] and Blecher and Smith [5] which characterise the normal completely bounded maps on the space of all bounded operators on a Hilbert space, which are module maps over a von Neumann algebra. We make great use of the weak* Haagerup tensor product of two von Neumann algebras, which is

Ideals with bounded approximate identities in Fourier algebras

Abstract 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.

Operator weak amenability of the Fourier algebra

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

Completely isometric representations of McbA(G) and UCB(Ĝ)ß

Let G be a locally compact group. It is shown that there exists a natural completely isometric representation of the completely bounded Fourier multiplier algebra M cb A(G), which is dual to the representation of the measure algebra M(G), on B(L 2 (G)). The image algebras of M(G) and M cb A(G) in CB σ (B(L 2 (G))) are intrinsically characterized, and some commutant theorems are proved. It is also shown that for any amenable group G, there is a natural completely isometric representation of UCB(G)* on B(L 2 (G)), which can be regarded as a duality result of Neufangs completely isometric representation theorem for LUC(G)*.

Operator amenability of Fourier–Stieltjes algebras

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

Spectral Synthesis and Operator Synthesis for Compact Groups

G

Biflatness and Pseudo-Amenability of Segal Algebras

is compact.

Operator Amenability of the Fourier Algebra in the cb-Multiplier Norm

LetG be a compact group andC(G )b e theC-algebra of continuous complex-valued functions onG. The paper constructs an imbedding of the Fourier algebra A(G )o fG into the algebra V(G )= C(G) h C(G) (Haagerup tensor product) and deduces results about parallel spectral synthesis, generalizing a result of Varopoulos. It then characterizes which diagonal sets in GG are sets of operator synthesis with respect to the Haar measure, using the denition of operator synthesis due to Arveson. This result is applied to obtain an analogue of a result of Froelich: a tensor formula for the algebras associated with the pre-orders dened by closed unital subsemigroups of G.

Operator biflatness of the Fourier algebra and approximate indicators for subgroups

We investigate generalized amenability and biflatness properties of various (operator) Segal algebras in both the group algebra, L1(G), and the Fourier algebra, A(G), of a locally compact group G. Received by the editors January 9, 2008. Published electronically May 20, 2010. The first author’s research was supported by an NSERC Post Doctoral Fellowship. The second author’s research was supported by NSERC under grant no. 312515-05. The third author’s research was supported by NSERC under grant no. 298444-04. AMS subject classification: 43A20, 43A30, 46H25, 46H10, 46H20, 46L07.

Completely isometric representations of _{}() and ()*

Let