Unbounded derivations in Bunce-Deddens-Toeplitz algebras (Q2633683)

Given an increasing sequence $\{l_k\}_{k=0}^{\infty}$ of nonnegative integers such that $l_k$ divides $l_{k+1}$ for $k\geq 0$, the Bunce-Deddens-Toeplitz algebra is defined as the $C^*$-algebra of operators on $l^2(\mathbb{Z}_{\geq 0})$, generated by all $l_k$-periodic weighted shifts for all $k\geq 0$. Different sequences $\{l_k\}$ may lead to the same algebras, with the classifying invariant being the supernatural number $N=\prod_{p-\text{prime}} p^{\varepsilon_p}$, where $\varepsilon_p=\sup \{j: \exists k\,\,\, p_j|l_k\}$. The authors adopt a slightly different definition of the Bunce-Deddens-Toeplitz algebra $A(N)$ associated with the supernatural number $N$ that uses $N$ more directly. They consider both finite and infinite $N$. The algebra $\mathcal{K}$ of compact operators on $l^2(\mathbb{Z}_{\geq 0})$ is contained in $A(N)$ and the quotient $\frac{A(N)}{\mathcal{K}}=B(N)$ is known as the Bunce-Deddens algebra. The structure of all those algebras is quite different, depending on whether $N$ is finite or infinite. The main objects of study in this paper are densely defined derivations $d : \mathcal{A}(N) \to A(N)$ in the Bunce-Deddens-Toeplitz algebras, where $\mathcal{A}(N)$ is the subalgebra of polynomials of $l_k$-periodic weighted shifts, as well as derivations $\delta : \mathcal{B}(N) \to B(N)$ in the Bunce-Deddens algebras, where $ \mathcal{B}(N)$ is the image of $\mathcal{A}(N)$ under the quotient map $A(N) \to \frac{A(N)}{\mathcal{K}}=B(N)$. Intriguingly, if $d : A(N) \to A(N)$ is a derivation, then $d$ preserves the ideal of compact operators. The main results of this paper are that any derivation in Bunce-Deddens or Bunce-Deddens-Toeplitz algebras can be uniquely decomposed into a sum of a certain special derivation and an approximately inner derivation. The special derivations are not approximately inner, they are explicitly described.