### Abstract

We initiate the study of the completely bounded multipliers
of the Haagerup tensor product A(G)⊗hA(G) of two copies of the Fourier
algebra A(G) of a locally compact group G. If E is a closed subset of
G we let E
♯ = {(s, t) : st ∈ E} and show that if E
♯
is a set of spectral
synthesis for A(G) ⊗h A(G) then E is a set of local spectral synthesis
for A(G). Conversely, 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.

Original language | English |
---|---|

Pages (from-to) | 1-40 |

Journal | International Journal of Mathematics |

Volume | 28 |

Issue number | 10 |

DOIs | |

Publication status | Published - 27 Jul 2017 |

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## Cite this

Alaghmandan, M., Todorov, I. G., & Turowska, L. (2017). Completely bounded bimodule maps and spectral synthesis.

*International Journal of Mathematics*,*28*(10), 1-40. https://doi.org/10.1142/S0129167X17500677