Closed ideals of $A^{\infty }$ and a famous problem of Grothendieck

Abstract

Using Fréchet algebraic technique, we show the existence of a nuclear Fréchet space without basis, thus providing yet another proof (of a different flavor) of a negative answer to a well known problem of Grothendieck from 1955. Using Fefferman's construction (which is based on complex-variable technique) of a $C^{\infty }$-function on the unit circle with certain properties, we give much simpler, transparent, and "natural" examples of restriction spaces without bases of nuclear Fréchet spaces of $C^{\infty }$-functions; these latter spaces, being classical objects of study, have attracted some attention because of their relevance to the theories of PDE and complex dynamical systems, and harmonic analysis. In particular, the restriction space $A^{\infty }(E)$, being a quotient algebra of the algebra $A^{\infty }(\Gamma )$, is the central one to other examples; the algebras $A^{\infty }$ had played a crucial role in solving a well-known problem of Kahane and Katznelson in the negative.