Diameter two properties in James spaces

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We study the diameter two properties in the spaces JH, JT∞ and JH∞. We show that the topological dual space of the previous Banach spaces fails every diameter two property. However, we prove that JH and JH∞ satisfy the strong diameter two property, and so the dual norm of these spaces is octahedral. Also we find a closed hyperplane M of JH∞ whose topological dual space enjoys the w*-strong diameter two property and also M and M* have an octahedral norm.