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Let H be a compact subgroup of a locally compact group G. We first investigate some (operator) (co)homological properties of the Fourier algebra A(G/H) of the homogeneous space G/H such as (operator) approximate biprojectivity and pseudo-contractibility. In particular, we show that A(G/H) is operator approximately biprojective if and only if G/H is discrete. We also show that A(G/H)^** is boundedly approximately amenable if and only if G is compact and H is open. Finally, we consider the question of existence of weakly compact multipliers on A(G/H).
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