Operator algebras associated with multiplicative convolutions of arithmetic functions

The action of N on l^2(N) is studied in association with the multiplicative structure of N. Then the maximal ideal space of the Banach algebra generated by N is homeomorphic to the product of closed unit disks indexed by primes, which reflects the fundamental theorem of arithmetic. The C*-algebra generated by 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 N is B(l^2(N)), the set of all bounded operators on l^2(N).Moreover, the differential operator on l^2(N,1/n(n+1)) defined by ▽f = μ * f is considered, where μ is the Mbius function. It is shown that the spectrum σ(▽) contains the closure of {ζ(s)-1: Re(s) > 1}. Interesting problems concerning are discussed.