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We used the Layer Korringa Kohn Rostoker technique to calculate the electronic structure of cobalt‐copper multilayers and spin valves from first principles within the local spin density approximation. Using this electronic structure together with a phenomenological self‐energy which may vary from layer to layer, we calculated the non‐local layer‐dependent conductivity by means of the Kubo linear response formalism. By calculating the majority and minority conductivities for parallel and anti‐parallel alignment of the moments in the cobalt layers we determined the giant magnetoresistance (GMR). Several interesting features emerge from the calculations. When the scattering rates are relatively high, we find that the contributions to the GMR are largely non‐local, with the largest contributions arising from changes in the currents carried in a cobalt plane next to copper due to fields sensed in the cobalt layer on the other side of copper. When scattering rates are relatively low (comparable to that of cobalt and copper at room temperature), there are important contributions to the GMR from local conduction in the copper layers. This effect arises from the fact that when the component of the majority spin electron momentum parallel to the layers exceeds a certain value, it gets trapped in the copper layers. If the scattering rate is lower in the copper than in the cobalt there is a significant enhancement in the majority spin conductivity and in the GMR. This effect is analogous to the channeling of light by an optical waveguide. © 1996 American Institute of Physics.