Operator biflatness of the Fourier algebra and approximate indicators for subgroups

Abstract

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 G = SL(3, C), the diagonal in G × G fails to have an approximate indicator.

On the operator homology of the Fourier algebra and its cb-multiplier completion

“…As applications of our techniques, we establish a hereditary property of inner amenability, answer an open question of Lau and Paterson [25], 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 ∆ 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.…”

“…The relative operator biflatness of A(G) has been studied by Ruan and Xu [35] and Aristov, Runde, and Spronk [3], where it was shown (by different methods) that A(G) is relatively operator biflat whenever G is QSIN, meaning L 1 (G) has a quasi-central bounded approximate identity (see [3,27,38]). The approach of Aristov, Runde, and Spronk is via approximate indicators, where they show that A(G) is relatively operator C-biflat whenever the diagonal subgroup G ∆ ≤ G × G has a bounded approximate indicator in B(G × G) of norm at most C. One of the main results of this paper establishes the converse when C = 1, that is, A(G) is relatively operator 1-biflat if and only if G ∆ has a contractive approximate indicator in B(G × G).…”

Inner amenability and approximation properties of locally compact quantum groups

“…The corresponding characterization for (relative) C-biflatness remains an interesting open question. In the co-commutative setting, the relative 1-biflatness of A(G) has been studied in [4,16,56]. It is known to be equivalent to the existence of a contractive approximate indicator for the diagonal subgroup G ∆ [16,Theorem 4.1].…”

Weak separation properties for closed subgroups of locally compact groups

“…For a closed subgroup H of G, the assertion that χ H is approximable may be viewed as a very weak form of subgroup separation. In [1], the discretized Fourier-Stieltjes algebra…”

“…Following the influential work of Ruan [34], much work on the homology of the Fourier algebra A (G) as a completely contractive Banach algebra has affirmed this as the appropriate category in which to consider A (G) and the related algebras of abstract harmonic analysis (e.g. [1,9,17,18]). Motivated by the success of this perspective, we focus on completely bounded projections, operator amenability, and the completely bounded multiplier algebra of A (G).…”