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

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 ->infinity p(n)1n=infinity, the Banach algebra A(p) is the set of all entire functions f such that f(z)=& sum;infinity n=0f(n)zn(z is an element of C),, where the sequence (f(n))n >= 0 of Taylor coefficients of f satisfies f(n)=O(p(n)-1)forn ->infinity with pointwise addition and scalar multiplication, a weighted Hadamard multiplication * with weight given byp(i.e.,(f & lowast;g)(z)=& sum;infinity n=0p(n)f(n)g(n)znforallz is an element of C), and the norm & Vert;f & Vert;=supn >= 0p(n)|f(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. center dot 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 Lech 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.(c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).