Spectral Synthesis and Operator Synthesis for Compact Groups

Let G be a compact group and C( G ) be the C * ‐algebra of continuous complex‐valued functions on G . The paper constructs an imbedding of the Fourier algebra A( G ) of G into the algebra V( G ) = C( G )⊗ h C( G ) (Haagerup tensor product) and deduces results about parallel spectral synthesis, generalizing a result of Varopoulos. It then characterizes which diagonal sets in G × G are sets of operator synthesis with respect to the Haar measure, using the definition of operator synthesis due to Arveson. This result is applied to obtain an analogue of a result of Froelich: a tensor formula for the algebras associated with the pre‐orders defined by closed unital subsemigroups of G .