Nontrivially Noetherian $C^*$-algebras

  • Taylor Hines
  • Erik Walsberg


We say that a $C^*$-algebra is Noetherian if it satisfies the ascending chain condition for two-sided closed ideals. A nontrivially Noetherian $C^*$-algebra is one with infinitely many ideals. Here, we show that nontrivially Noetherian $C^*$-algebras exist, and that a separable $C^*$-algebra is Noetherian if and only if it contains countably many ideals and has no infinite strictly ascending chain of primitive ideals. Furthermore, we prove that every Noetherian $C^*$-algebra has a finite-dimensional center. Where possible, we extend results about the ideal structure of $C^*$-algebras to Artinian $C^*$-algebras (those satisfying the descending chain condition for closed ideals).
How to Cite
Hines, T., & Walsberg, E. (2012). Nontrivially Noetherian $C^*$-algebras. MATHEMATICA SCANDINAVICA, 111(1), 135-146.