Operator algebras associated with multiplicative convolutions of arithmetic functions

• 来源期刊: 中国科学（数学）
• 年，卷（期）: 2018,（9）
• 页码范围: 1665-1676
• 页数: 12页
• 语种: 中文

摘要

The action of N on l2(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(l2(N)),the set of all bounded operators on l2(N).Moreover,the differential operator on l2(N,1/n(n+1) defined by ▽f =μ* f is considered,where μ is the M(o)bius function.It is shown that the spectrum σ(▽) contains the closure of {ζ(s)-1:Re(s) ＞ 1}.Interesting problems concerning ▽ are discussed.