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The purpose of this paper is to study the relation between the $C^0$-topologyand the topology induced by the spectral norm on the group of Hamiltoniandiffeomorphisms of a closed symplectic manifold. Following the approach ofBuhovsky-Humili\`ere-Seyfaddini, we prove the $C^0$-continuity of the spectralnorm for complex projective spaces and negative monotone symplectic manifolds.The case of complex projective spaces provides an alternative approach to the$C^0$-continuity of the spectral norm proven by Shelukhin. We also prove apartial $C^0$-continuity of the spectral norm for rational symplecticmanifolds. Some applications such as the Arnold conjecture in the context of$C^0$-symplectic topology are also discussed.

On $C^0$-continuity of the spectral norm for symplectically non-aspherical manifolds