Determinants and traces of multidimensional discrete periodic operators with defects

As it is shown in previous works, discrete periodic operators with defects are unitarily equivalent to the operators of the form $${\mathcal A}{\bf u}={\bf A}_0{\bf u}+{\bf A}_1\int_0^1dk_1{\bf B}_1{\bf u}+...+{\bf A}_N\int_0^1dk_1...\int_0^1dk_N{\bf B}_N{\bf u},\ \ {\bf u}\in L^2([0,1]^N,\mathbb{C}^M),$$ where $({\bf A},{\bf B})(k_1,...,k_N)$ are continuous matrix-valued functions of appropriate sizes. All such operators form a non-closed algebra ${\mathscr H}_{N,M}$. In this article we show that there exist a trace $\pmb{\tau}$ and a determinant $\pmb{\pi}$ defined for operators from ${\mathscr H}_{N,M}$ with the properties $$\pmb{\tau}(\alpha{\mathcal A}+\beta{\mathcal B})=\alpha\pmb{\tau}({\mathcal A})+\beta\pmb{\tau}({\mathcal B}),\ \ \pmb{\tau}({\mathcal A}{\mathcal B})=\pmb{\tau}({\mathcal B}{\mathcal A}),\ \ \pmb{\pi}({\mathcal A}{\mathcal B})=\pmb{\pi}({\mathcal A})\pmb{\pi}({\mathcal B}),\ \ \pmb{\pi}(e^{{\mathcal A}})=e^{\pmb{\tau}({\mathcal A})}.$$ The mappings $\pmb{\pi}$, $\pmb{\tau}$ are vector-valued functions. While $\pmb{\pi}$ has a complex structure, $\pmb{\tau}$ is simple $$\pmb{\tau}({\mathcal A})=\left({\rm Tr}{\bf A}_0,\int_0^1dk_1{\rm Tr}{\bf B}_1{\bf A}_1,...,\int_0^1dk_1...\int_0^1dk_N{\rm Tr}{\bf B}_N{\bf A}_N\right).$$ There exists the norm under which the closure $\overline{{\mathscr H}}_{N,M}$ is a Banach algebra, and $\pmb{\pi}$, $\pmb{\tau}$ are continuous (analytic) mappings. This algebra contains simultaneously all operators of multiplication by matrix-valued functions and all operators from the trace class. Thus, it generalizes the other algebras for which determinants and traces was previously defined.