Closed ideals of \(A^\infty\) and a famous problem of Grothendieck (Q643334)

In the paper the author constructs another examples of nuclear Fréchet spaces without bases. The Fréchet algebra techniques play an important role. The paper consists of four sections. In the introductory part the author gives a brief history of the problem. The second part recalls basic definitions and well-known facts. The main result is contained in Theorem 2.7. It establishes examples of nuclear Fréchet spaces without a basis which are quotients of some Fréchet algebras. In Section 4 the author gives more concrete examples of nuclear Fréchet spaces without a basis. The first of them deals with the space \(A^{\infty}(\mathbb{T})\) and \(I(E):=\{f\in A^{\infty}(\mathbb{T}):\,\,f\equiv0\,\,\text{on}\,\,E\}\) for a closed, disconnected infinite \(E\subset\mathbb{T}\) such that \(I(E)\) is a proper, closed ideal of \(A^{\infty}(\mathbb{T})\). Then in Section 3 it is shown that for some \(E\) the ideal \(I(E)\) is indeed proper.