Quantum scalar field on the massless $\mathrm{}(2+1)\mathrm{}$-dimensional black hole background

The behavior of a quantum scalar field is studied in the metric ground state of the (2+1)-dimensional black hole of Bañados, Teitelboim, and Zanelli which contains a naked singularity. The one-loop BTZ partition function, the associate black hole effective entropy, and the expectation value of the quantum fluctuation, as well as the renormalized expectation value of the stress tensor are explicitly computed in the framework of the $\zeta$-function procedure. This is done for all values of the coupling with the curvature, the mass of the field, and the temperature of the quantum state. In the massless conformally coupled case, the found stress tensor is used for determining the quantum back reaction on the metric due to the scalar field in the quantum vacuum state, by solving the semiclassical Einstein equations. It is finally argued that, within the framework of the $1/N$ expansion, the cosmic censorship hypothesis is implemented since the naked singularity of the ground state metric is shielded by an event horizon created by the back reaction.