We will discuss our construction of a non-diagonalizable pure state of the C*-algebra of all bounded linear operators on the separable Hilbert space. It solved Anderson's conjecture in ZFC. Previous examples of such pure states were obtained under additional set-theoretic assumptions. By the positive solution to the Kadison-Singer problem the pure state constructed is also an example of a pure state whose restriction to any atomic masa is not multiplicative. This is related to a construction under the continuum hypothesis due to Akemann and Weaver of a pure state whose restriction to any masa is not multiplicative.