I see 3 viable choices: Subjective, objective, and attributive (or “descriptive”) genitive. The latter choice would seem at first blush to flow most logically from the preceding passage, where James has just propounded his view of “true religion”–an attributive genitive would now seem to be establishing guidelines for how [not] to “practice the Christian faith.”

An “objective genitive, on the other hand, would seem to set up James’ subsequent use of “faith” in the rest of chap. 2, where the faith of a given “brother” among those being addressed by James can actually be “dead.” This could also be made to fit the preceding passage by reading “your faith in Christ” for the articular pistis.

A subjective genitive would put an entirely different spin on the passage, where it would seem that James is commanding his readers to be circumspect about how they “appropriate” the faithfulness of Jesus, along the lines of what Bobby is saying with respect to TFT. This would seem to have echoes of “participation” or “union” with Christ in his faithfulness to the “least of these” among the brethren, but that reading of pistis would almost certainly be entirely distinct from the subsequent mention of “faith” in the same chapter.

Here’s a post highlighting how TFT thought of this, and how Dan Wallace approaches this (in a more guarded but curious way):

http://evangelicalcalvinist.com/2010/01/06/some-greek-the-faith-of-christ-in-galatians-220/