Almost periodic factorization in the twenty first century

The set $AP$ of (Bohr) almost periodic functions is the closed subalgebra of $L_\infty(\mathbb R)$ generated by all the exponents $e_\lambda(x):=e^{i\lambda x}$, $\lambda\in\mathbb R$. An $AP$ factorization of an $n$-by-$n$ matrix function $G$ is its representation as a product \[ G=G_+\text{diag}[e_{\lambda_1},\ldots,e_{\lambda_n}]G_-,\] where $G_+^{\pm 1}$ and $G_-^{\pm 1}$ have all entries in $AP$ with non-negative (resp., non-positive) Bohr-Fourier coefficients. This is a natural generalization of the classical Wiener-Hopf factorization of continuous matrix-functions on the unit circle, arising in particular when considering convolution type equation on finite intervals. The state of $AP$ factorization theory as of the beginning of the century can be found in the monograph ``Convolution Operators and Factorization of Almost Periodic Matrix Functions\u27\u27 by A. B\ ottcher, Yu. I. Karlovich and myself. In this talk I will describe, time permitting, further results obtained and problems still open