On a Banach algebra of entire functions with a weighted Hadamard multiplication

Abstract

New algebraic-analytic properties of a previously studied Banach algebra A(p) of entire functions are established. For a given fixed sequence (p(n))n≥0 of positive real numbers, such that limn→∞⁡p(n)1n=∞, the Banach algebra A(p) is the set of all entire functions f such that f(z)=∑n=0∞fˆ(n)zn (z∈C), where the sequence (fˆ(n))n≥0 of Taylor coefficients of f satisfies fˆ(n)=O(p(n)−1) for n→∞, with pointwise addition and scalar multiplication, a weighted Hadamard multiplication ⁎ with weight given by p (i.e., (f⁎g)(z)=∑n=0∞p(n)fˆ(n)gˆ(n)zn for all z∈C), and the norm ‖f‖=supn≥0⁡p(n)|fˆ(n)|. The following results are shown:•The Topological stable rank of A(p) is 1.•The Bass stable rank of A(p) is 1.•A(p) is a Hermite ring.•A(p) is not a projective-free ring.•Idempotents in A(p) are described.•Exponentials in A(p) are described, and it is shown that every invertible element of A(p) has a logarithm, so that the first Čech cohomology group H1(M(A(p)),Z) with integer coefficients of the maximal ideal space M(A(p)) is trivial.•A generalised necessary and sufficient ‘corona-type condition’ on the matricial data (A,b) with entries from A(p) is given for the solvability of Ax=b with x also having entries from A(p).•The Krull dimension of A(p) is infinite.•A(p) is neither Artinian nor Noetherian.•A(p) is coherent.•The special linear group over A(p) is generated by elementary matrices.