I would like to thank Prof. Richard M. Timoney for the care he took in supervising my PhD thesis.

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It is well known that every derivation of a von Neumann algebra into itself is an inner derivation and that every derivation of a von Neumann algebra into its predual is inner. It is less well known that every triple derivation (defined below) of a von Neumann algebra into itself is an inner triple derivation.We examine to what extent all triple derivations of a von Neumann algebra into its predual are inner. This rarely happens but it comes close. We prove a (triple) cohomological characterization of finite factors and a zero-one law for factors.2010 Mathematics Subject Classification: Primary 46L57, 17C65; Secondary 46L10. Key words and phrases: triple derivation, ternary weak amenability, von Neumann factor, commutator.( 1 ) In this paper, we use the terms 'triple' and 'ternary' interchangeably, while mindful that in some quarters 'triple' means 'Jordan triple' and 'ternary' refers to an associative triple setting, such as a TRO (ternary ring of operators).

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To each projection p in a C * -algebra A we associate a family of derivations on A, called p-derivations, and relate them to the space of triple derivations on pA(1 − p). We then show that every derivation on a ternary ring of operators is spatial and we investigate whether every such derivation on a weakly closed ternary ring of operators is inner.