The supernova SN 2005cz has recently attracted some attention, due to the fact that it was spectroscopically similar to type Ib supernovae (SNe), a class that is presumed to result from core-collapse of massive stars, yet it occurred in an elliptical galaxy, where one expects very few massive stars to exist. Two explanations for this remarkable event were put forward. Perets et al. (2010) associate SN 2005cz with the class of Ca-rich, faint SNe Ib, which likely result from old double-white-dwarf systems with a He-rich secondary. On the other hand, Kawabata et al. (2010) suggest that SN 2005cz is indeed a core-collapse event (in a binary system), albeit of a star at the lower end of the mass range, 10-12 M_Sun. The existence of this star in its elliptical host is explained as resulting from low-level star formation (SF) activity in that galaxy. Here we present extensive observations of the location of SN 2005cz, sensitive to a variety of SF tracers, including optical spectroscopy, H_alpha emission, UV emission and HST photometry. We show that NGC 4589, the host galaxy of SN 2005cz, does not show any signatures of a young stellar population or recent SF activity either close to or far from the location of SN 2005cz.