A simple mesoscopic structure consisting of a double symmetric loops coupled by a segment of length d0 in the presence of an Aharonov-Bohm flux is designed to obtain transmission band gaps and Fano resonances. A general analytical expression for the transmission coefficient and the density of states (DOS) are obtained for various systems of this kind within the framework of the Green's function method in the presence of the magnetic flux. In this work, the amplitude of the transmission and DOS are discussed as a function of the wave vector. We show that the transmission spectrum of the whole structure may exhibit a band gap and a resonance of Fano type without introducing any impurity in one arm of the loop. In particular, we show that for specific values of the magnetic flux and the lengths of the arms constituting the loops, the Fano resonance collapses giving rise to the so-called trapped states or bound in continuum (BIC) states. These states appear when the width of the Fano resonance vanishes in the transmission coefficient as well as in the density of states. Also, we show that the shape of the Fano resonances and the width of the band gaps are very sensitive to the value of the magnetic flux and the geometry of the structure. These results may have important applications for electronic transport in mesoscopic systems.