Completely bounded maps and invariant subspaces

Abstract

We provide a description of certain invariance properties of completely bounded bimodule maps in terms of their symbols. If G is a locally compact quantum group, we characterise the completely bounded L ∞ (G) ′ -bimodule maps that send C0(Ĝ) into L ∞ (Ĝ) in terms of the properties of the corresponding elements of the normal Haagerup tensor product L ∞ (G) ⊗σ h L ∞ (G). As a consequence, we obtain an intrinsic characterisation of the normal completely bounded L ∞ (G) ′ -bimodule maps that leave L ∞ (Ĝ) invariant, extending and unifying results, formulated in the current literature separately for the commutative and the co-commutative cases.

Reduced spectral synthesis and compact operator synthesis

“…, d π } (or i, j ∈ N if d π = ∞). Moreover, by [3,Lemma 3.10],h π i,j ∈ S(G, G); thus, as, for any r ∈ G, we have thath π i,j (sr, tr) =h π i,j (s, t) for almost all (s, t), h π i,j = N (u π i,j ) for some element u π i,j ∈ M cb A(G), vanishing on E [1]. By Claim 2, Shπ i,j (T K ) = P (K)Shπ i,j…”