Completely bounded bimodule maps and spectral synthesis

• We initiate the study of the completely bounded multipliers of the Haagerup tensor product A(G) circle times(h) A(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) circle times(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)circle times(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)circle times(h) A(G), we show that, in the case G is weakly amenable, such a map leaves the multiplication algebra of L-infinity(G) invariant if and only if its support is contained in the antidiagonal of G.

Ämnesord

NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

Nyckelord

Fourier algebra

completely bounded map

bimodule

operator space

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art (ämneskategori)

ref (ämneskategori)