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In this paper we show that in the presence of an anti-symmetric tensor$B$-background, Witten's star algebra for open string fields persists topossess the structure of a direct product of commuting Moyal pairs. Theinterplay between the noncommutativity due to three-string overlap and that dueto the $B$-background is our main concern. In each pair of noncommutativedirections parallel to the $B$-background, the Moyal pairs mix string modes inthe two directions and are labeled, in addition to a continuous parameter, by{\it two} discrete values as well. However, the Moyal parameters are$B$-dependent only for discrete pairs. We have also demonstrated the large-$B$contraction of the star algebra, with one of the discrete Moyal pairs droppingout while the other giving rise to the center-of-mass noncommutative functionalgebra.Comment: minor notation chang

Moyal representation of the string field star product in the presence of aBbackground