Cones and norms in the tensor product of matrix spaces

This paper is of expository nature. In the tensor product of two matrix spaces and we consider three natural cones; the first is the cone of positive semi-definite matrices, the second is the cone generated by tensor products of positive semi-definite matrices in the factor spaces, and the third cone is defined as the dual cone of . Correspondingly, in the tensor space identified with , besides the spectral and the trace norms, we consider several tensor norms generated by sepectral and/or trace norms in the factor spaces. We will study relations among those norms as well as relations among the three kinds of order intervals in the tensor product.