We list some open problems in C\(^{*}\)-algebra:

Question 1 (Naimark's Problem):

Every C\(^{*}\)-algebra that has only one irreducible \(*\)-representation up to unitary equivalence is isomorphic to the \(*\)-algebra of compact operators on some (not necessarily separable) Hilbert space.

Whether Naimark problem is independent of \(\mathsf{ZFC}\)?

Akemann & Weaver (2004)1 used the \(\diamondsuit\)-Principle to construct a C\(^{*}\)-algebra with \(\aleph _{1}\) generators that serves as a counterexample to Naimark's Problem. But in general its consistency remains unknown.

Question 2:

Every C\(^{*}\)-algebras of density of continuum \(\aleph_{1} = 2^{\aleph_{0}}\) has no nonseparable commutative subalgebra.2

Akemann & Donner (1979) constructs example with a C\(^{*}\)-algebra with \(\aleph _{1}\) generators with only separable abelian C\(^{*}\)-subalgebras. Bice and Koszmider (2017)3 remove their assumption of the continuum hypothesis. Whether

Question 3:

Every amenable operator algebra is isomorphic to a (necessarily nuclear) C\(^{*}\)-algebra.

Resolved. Choi, Farah, & Ozawa (2013) 4 construct a counterexample to this problem in non-separable case, whereas separable case remains open.

Question 4:

Do there exist non-discrete second countable locally compact groups which are C\(^{*}\)-simple?5

Resolved. Suzuki (2017).6

Reference

[1] Naimark, M. A. (1948), "Rings with involutions", Uspekhi Mat. Nauk, 3: 52-145

[2] Naimark, M. A. (1951), "On a problem in the theory of rings with involution", Uspekhi Mat. Nauk, 6: 160-164

  1. Akemann, Charles; Weaver, Nik (2004), "Consistency of a counterexample to Naimark's problem", Proceedings of the National Academy of Sciences of the United States of America, 101 (20): 7522–7525↩︎

  2. C. Akemann and J. Doner, A nonseparable C\(^{*}\)-algebra with only separable abelian C\(^{*}\)-subalgebras. Bull. London Math. Soc. 11 (1979), no. 3, 279–284.↩︎

  3. Bice, T., & Koszmider, P. (2017). A note on the Akemann-Doner and Farah-Wofsey constructions. Proceedings of the American Mathematical Society, 145(2), 681-687.↩︎

  4. Choi, Y., Farah, I., & Ozawa, N. (2014, February). A NONSEPARABLE AMENABLE OPERATOR ALGEBRA WHICH IS NOT ISOMORPHIC TO A-ALGEBRA. In Forum of Mathematics, Sigma (Vol. 2, p. e2). Cambridge University Press.↩︎

  5. Pierre de la Harpe, On simplicity of reduced C\(^{*}\)-algebras of groups, Bull. Lond. Math. Soc.↩︎

  6. Suzuki, Y. (2017). Elementary constructions of non-discrete 𝐶*-simple groups. Proceedings of the American Mathematical Society, 145(3), 1369-1371.↩︎