The composition of a planet's atmosphere has to do with the RMS velocity of gas molecules at a given planetary temperature. When this velocity exceeds the escape velocity of the planet, that gas is lost to space.
But there is one more factor. In the absence of a magnetic field, gas molecules can dissociate from being hit with the particles from the solar wind. E.g., water can dissociate into oxygen and hydrogen, and hydrogen having a relatively high RMS velocity readily leaks out to space. The remaining oxygen is too reactive to remain and then forms carbonates in rocks and carbon dioxide in the atmosphere. This is, from what I read, the explanation for the atmospheres of both Mars and Venus, which have only a small to non-existent magnetic field.
So yes, a magnetic field seems to be essential to holding a life-friendly atmosphere.
But there is one more factor. In the absence of a magnetic field, gas molecules can dissociate from being hit with the particles from the solar wind. E.g., water can dissociate into oxygen and hydrogen, and hydrogen having a relatively high RMS velocity readily leaks out to space. The remaining oxygen is too reactive to remain and then forms carbonates in rocks and carbon dioxide in the atmosphere. This is, from what I read, the explanation for the atmospheres of both Mars and Venus, which have only a small to non-existent magnetic field.
So yes, a magnetic field seems to be essential to holding a life-friendly atmosphere.