Weak spectral synthesis in commutative Banach algebras. II

Abstract

Let A be a semisimple and regular commutative Banach algebra with structure space Δ ( A ) . Continuing our investigation in [E. Kaniuth, Weak spectral synthesis in commutative Banach algebras, J. Funct. Anal. 254 (2008) 987–1002], we establish various results on intersections and unions of weak spectral sets and weak Ditkin sets in Δ ( A ) . As an important example, the algebra of n-times continuously differentiable functions is studied in detail. In addition, we prove a theorem on spectral synthesis for projective tensor products of commutative Banach algebras which applies to Fourier algebras of locally compact groups.