Completely bounded bimodule maps and spectral synthesis

Abstract

We initiate the study of the completely bounded multipliers of the Haagerup tensor product A(G)⊗ h A(G) of two copies of the Fourier algebra A(G) of a locally compact group G. If E is a closed subset ofConversely, we prove that if E is a set of spectral synthesis for A(G) and G is a Moore group then E ♯ is a set of spectral synthesis for A(G) ⊗ h A(G). Using the natural identification of the space of all completely bounded weak* continuous VN(G) ′ -bimodule maps with the dual of A(G) ⊗ h A(G), we show that, in the case G is weakly amenable, such a map leaves the multiplication algebra of L ∞ (G) invariant if and only if its support is contained in the antidiagonal of G.

Completely bounded maps and invariant subspaces

“…Specialising to the case of co-commutative quantum groups, we obtain a generalisation of our previous results from [1], removing the assumption of weak amenability imposed therein. On the other hand, specialising to the case of commutative quantum groups, we obtain the first, to the best of our knowledge, rigorous characterisation of the Schur multipliers ϕ : G × G → C whose corresponding map on B(L 2 (G)) leaves VN(G) invariant, in terms of the function ϕ alone; the result states that this happens if and only if, for every r ∈ G, the equality ϕ(sr, tr) = ϕ(s, t) holds for marginally almost all (s, t) ∈ G × G.…”

“…In [1], A eh (G) (resp. A h (G)) was identified with a completely contractive Banach algebra of separately (resp.…”

“…Note that A h (G) is a regular (semi-simple) Banach algebra whose Gelfand spectrum can be homeomorphically identified with G × G. M. Daws has shown [8] that, if f ∈ A(G) then m * (f ) belongs to the algebra M cb A h (G) of completely bounded multipliers [1] of A h (G); in particular,…”

“…Since VN(G) ⊗ eh VN(G) naturally embeds in VN(G × G), to each such map there corresponds canonically an element of the Neumann algebra VN(G × G). Those such maps that leave L ∞ (G) invariant were characterised by M. Neufang [19] (see also [20]) as arising in a canonical fashion from bounded complex measures on G. In [1], under the restriction that G be weakly amenable, we provided an intrinsic characterisation of the symbols from VN(G) ⊗ eh VN(G) whose maps leave L ∞ (G) invariant, showing that, when viewed as elements of VN(G × G), they are precisely those supported, in the sense of P. Eymard [11], on the anti-diagonal of G.…”

Mapping ideals of quantum group multipliers

“…As further corollaries, we (a) deduce a spectral synthesis result for the bivariate Fourier algebra A h (G) = A(G) ⊗ h A(G) (see [1]) of a QSIN group G, which, in particular, fills a gap in [46] (see Remark 4.6), and (b) show that L 1 (G) is completely isomorphic to an operator algebra if and only if G is finite. The isomorphism (2) also simultaneously generalizes, and provides a new proof of, the classical inclusion M cb A(G) ⊆ WAP(G) [76], where WAP(G) is the space of weakly almost periodic functions on G. This partially answers a question raised in [36,Remark 5.7].…”