Is it permissible to share the fare for a ride to work with a Christian woman, and how does one invite Christian women to Islam?

The preponderant view is that there is no harm in the two of you renting a shared car to transport you to work, provided that no khalwa (private seclusion) occurs between you. This is coupled with the obligation not to sit next to her and to lower your gaze from her. As for inviting her to Islam, it is preferable that you seek out a woman to do that. However, if you cannot find one, then it is permissible for you to invite her, while exercising extreme caution against khalwa or falling into anything unlawful.