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In this paper, we define the eta cochain form and prove its regularity whenthe kernel of a family of Dirac operators is a vector bundle. We decompose theeta form as a pairing of the eta cochain form with the Chern character of anidempotent matrix and we also decompose the Chern character of the index bundlefor a fibration with boundary as a pairing of the family Chern-Connes characterfor a manifold with boundary with the Chern character of an idempotent matrix.We define the family $b$-Chern-Connes character and then we prove that it isentire and give its variation formula. By this variation formula, we proveanother noncommutative family Atiyah-Patodi-Singer index theorem. Thus, weextend the results of Gezler and Wu to the family case.

The noncommutative family Atiyah-Patodi-Singer index theorem