Operator algebras associated with multiplicative convolutions of arithmetic functions

Abstract

The action of $\mathbb{N}$ on $l^2(\mathbb{N})$ is studied in association with the multiplicative structure of $\mathbb{N}$. Then the maximal ideal space of the Banach algebra generated by $\mathbb{N}$ is homeomorphic to the product of closed unit disks indexed by primes, which reflects the fundamental theorem of arithmetic. The $C^\ast$-algebra generated by $\mathbb{N}$ does not contain any non-zero projection of finite rank. This assertion is equivalent to the existence of infinitely many primes. The von Neumann algebra generated by $\mathbb{N}$ is $B(l^2(\mathbb{N}))$, the set of all bounded operators on $l^2(\mathbb{N})$. Moreover, the differential operator on $l^2(\mathbb{N},\frac{1}{n(n+1)})$ defined by $\nabla~f=\mu\ast~f$ is considered, where $\mu$ is the Mobius function. It is shown that the spectrum $\sigma(\nabla)$ contains the closure of $\{\zeta(s)^{-1}:~{\rm~Re}(s)>1\}$. Interesting problems concerning $\nabla$ are discussed.