On the Bezout equation in the ring of periodic distributions

A corona type theorem is given for the ring R of periodic distributions in R^d in terms of the sequence of Fourier coefficients of these distributions, which have at most polynomial growth. It is also shown that the Bass stable rank and the topological stable rank of R are both equal to 1.Comment: 10 page

Logarithms and exponentials in Banach algebras

Let $A$ be a complex Banach algebra. If the spectrum of an invertible element $a\in A$ does not separate the plane, then $a$ admits a logarithm. We present two elementary proofs of this classical result which are independent of the holomorphic functional calculus. We also discuss the case of real Banach algebras. As applications, we obtain simple proofs that every invertible matrix over $\mathbb C$ has a logarithm and that every real matrix $M$ in $M_n(\mathbb R)$ with $\det M>0$ is a product of two real exponential matrices.Comment: 9 page

Logarithms and Exponentials in the Matrix Algebra M2(A)

International audienceIt is well known that in the disk-algebra A(D) every zero-free function has a logarithm in A(D). This is no longer true if we look at invertible matrices over A(D). In this paper we give a sufficient condition on the trace of a 2 × 2-matrix M in order M = e L for some matrix L ∈ A(D). We compute all the logarithms of the identity matrix in M2(A(D)) and observe that the anti-diagonal elements can be arbitrarily prescribed. We also characterize those upper (or lower) triangular matrices which are exponentials in M2(A(D)) and determine all their logarithms. This will enable us to prove that exp M2(A(D)) is neither closed nor open within the principal component of M2(A(D)) −1. Finally, we show that every invertible matrix in M2(A(D)) is a product of four exponential matrices and give conditions for reducing this number. These results will be put into the more general setting of commutative Banach algebras whenever possible