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 lim n ∞ p(n)∞, with pointwise addition and scalar multiplication, a weighted Hadamard multiplication * with weight given by p (i.e., (f * g)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 H 1 (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.