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A detailed first-principles study of the atomic and electronic structure of the Co/Al2O3/Co magnetic tunnel junction has been performed in order to elucidate the key features determining the spin-dependent tunneling. The atomic structure of the multilayer with the O- and Al-terminated interfaces between fcc Co(111) and crystalline α-Al2O3(0001) has been optimized using self-consistent spin-polarized calculations within densityfunctional theory and the generalized gradient approximation. We found that the relaxed atomic structure of the O-terminated interface is characterized by a rippling of the Co interfacial plane, the average Co-O bond length being 2.04 Å which is within 5% of that in bulk CoO. The corresponding electronic structure is influenced by the covalent bonding between the O 2p and Co 3d orbitals resulting in exchange-split bonding and antibonding states and an induced magnetic moment of 0.07 µB on the interfacial oxygen atoms. The Al-terminated interface contains Co-Al bonds with an average bond length of 2.49 Å compared to 2.48 Å in bulk CoAl. Due to charge transfer and screening effects the Co interfacial layer acquires a negative charge which results in a reduced magnetic moment of 1.15 µB per Co atom. We found that the electronic structure of the O-terminated Co/Al2O3/Co tunnel junction exhibits negative spin polarization at the Fermi energy within the first few monolayers of alumina but it eventually becomes positive for distances beyond 10 Å.