We investigate the properties of the high-order discontinuous Galerkin spectral element method (DGSEM) with implicit backward-Euler time stepping for the approximation of scalar hyperbolic conservation laws in multiple space dimensions. We first prove that the DGSEM scheme in one space dimension preserves a maximum principle for the cell-averaged solution when the time step is large enough. This property however no longer holds in multiple space dimensions and we propose to use the flux-corrected transport limiting [Boris and Book, J. Comput. Phys., 11 (1973), 38-69] based on a low-order approximation using graph viscosity to impose a maximum principle on the cell-averaged solution. These results allow to use a linear scaling limiter [Zhang and Shu, J. Comput. Phys., 229 (2010), 3091-3120] in order to impose a maximum principle at all DOFs within elements. Numerical experiments in one and two space dimensions support the conclusions of the present analyses and highlight high-resolution of the limited scheme.