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The complex Gaussian multiple-access channel (GMAC) Y_=AX_+N_ is considered. The transmitters send information independently with a power constraint so that X_ has a product distribution with E[|X/sub k/|/sup 2/]/spl les/P/sub k/. It is known that multiuser codes exist that will achieve any rate-tuple in the capacity region of the GMAC provided that an optimum joint decoder is used. However, little progress has been made in multiuser coding, and moreover, optimum joint decoding would be too complex. We restrict the decoder to be of a successive decoding type with equalization. Such a decoder is parameterized by feedforward and feedback equalization vectors. The optimum successive decoder (OSD) is obtained by maximizing the mutual information for each user over those vectors. The key result of this paper is that the OSD achieves the total capacity of the GMAC at any vertex of the capacity region. With the OSD, each transmitter can use a single-user code independently of the other users. The complexity of equalization is also only linear in the number of users. For the conventional GMAC, Y=/spl Sigma//sub i=1//sup M/ X/sub i/+N, where A is a row vector with unit elements, the OSD is degenerate and involves no equalization. It reduces to the well-known successive decoder for that channel as described by Wynter (1974) and Cover (1975). Furthermore, for the particular case of the uncoded channel, the OSD reduces to a new decision feedback multiuser detector. This detector is optimum in the sense that it maximizes signal-to-interference ratio for each user.