  Our first main result states that the spectral norm on the group of
Hamiltonian diffeomorphisms, introduced in the works of Viterbo, Schwarz and
Oh, is continuous with respect to the C^0 topology, when M is symplectically
aspherical. This statement was previously proven only in the case of closed
surfaces. As a corollary, using a recent result of Kislev and Shelukhin, we
obtain C^0 continuity of barcodes on aspherical symplectic manifolds, and
furthermore define barcodes for Hamiltonian homeomorphisms. We also present
several applications to Hofer geometry and dynamics of Hamiltonian
homeomorphisms.
  Our second main result is related to the Arnold conjecture about fixed points
of Hamiltonian diffeomorphisms. The recent example of a Hamiltonian
homeomorphism, on any closed symplectic manifold of dimension greater than 2,
having only one fixed point, shows that the conjecture does not admit a direct
generalization to the C^0 setting. However, in this paper we demonstrate that a
reformulation of the conjecture in terms of fixed points as well as spectral
invariants still holds for Hamiltonian homeomorphisms on symplectically
aspherical manifolds.
