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We propose a non-data-aided adaptive beamforming algorithm based on Widely Linear (WL) processing techniques and the Auxiliary Vector Filtering (AVF) algorithm for non-circular signals, where only the steering vector of the desired user is known. The proposed Widely Linear Auxiliary Vector Filtering (WL-AVF) algorithm recursively updates the filter weights by a sequence of auxiliary vectors that are designed according to the Widely Linearly Constrained Minimum Variance (WLCMV) criterion. It takes full advantage of the second-order statistics of the non-circular data, achieving a higher maximum signal-to-interference-plus-noise ratio (SINR) than the linear AVF. Key properties of the proposed WL-AVF are analyzed. Simulation results show that the WL-AVF beamforming algorithm performs the best among the existing adaptive algorithms.