A trailing rope tensioning device for an elevator installation engages a trailing rope that extends from a first hitch point of a car frame to a second hitch point of a counterweight frame. The tensioning device has two tensioning pulleys that are arranged offset relative to a vertical direction of travel of the elevator and a transverse horizontal direction. This arrangement of the tensioning pulleys makes it possible to use relatively large tensioning pulleys resulting in a lower number of revolutions per unit of distance traveled, which brings about a reduction in noise from the bearings rotatably mounting the pulleys. Larger diameters of trailing rope and, as a result, fewer trailing ropes can be used. Furthermore, the tensioning pulleys can be standardized.