Moyal representation of the string field star product in the presence of a B background

In this paper we show that in the presence of an antisymmetric tensor B background, Witten's star algebra for open string fields persists to possess the structure of a direct product of commuting Moyal pairs. The interplay between the noncommutativity due to three-string overlap and that due to the B background is our main concern. In each pair of noncommutative directions parallel to the B background, the Moyal pairs mix string modes in the two directions and are labeled, in addition to a continuous parameter, by 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 dropping out and with the other giving rise to the center-of-mass noncommutative function algebra.