Elementary operators on Banach algebras and Fourier transform

Abstract

We consider elementary operators x → ∑ j=1n a j xb j , acting on a unital Banach algebra, where a j and b j are separately commuting families of generalized scalar elements. We give an ascent estimate and a lower bound estimate for such an operator. Additionally, we give a weak variant of the Fuglede-Putnam theorem for an elementary operator with strongly commuting families {a j } and {b j }, i.e. a j = a′ j + ia″ j (b j = b″ j +ib″ j ), where all a′ j and a″ j (b′ j and b″ j ) commute. The main tool is an L 1 estimate of the Fourier transform of a certain class of C cpt∞ functions on ℝ 2n .