We construct a pure state on the C*-algebra [Formula: see text] of all bounded linear operators on [Formula: see text], which is not diagonalizable [i.e., it is not of the form [Formula: see text] for any orthonormal basis [Formula: see text] of [Formula: see text] and an ultrafilter u on N]. This constitutes a counterexample to Anderson's conjecture without additional hypothesis and improves results of C. Akemann, N. Weaver, I. Farah, and I. Smythe who constructed such states making additional set-theoretic assumptions. It follows from results of J. Anderson and the positive solution to the Kadison-Singer problem due to A. Marcus, D. Spielman, and N. Srivastava that the restriction of our pure state to any atomic masa [Formula: see text] of diagonal operators with respect to an orthonormal basis [Formula: see text] is not multiplicative on [Formula: see text].
Keywords: Anderson’s conjecture; atomic masa; diagonal operators; pure state.