# Michael Dritschel, Newcastle University (UK): Completely bounded kernels

Abstract: The Kolmogorov decomposition of positive scalar valued kernels has played an important role in applications of operator theory to function theory. It has been vastly generalised, finding its apotheosis in the result of Baretto, Bhat, Liebscher and Skeide which states that a positive $L(A,B)$-valued kernel, $A$ and $B$ $C^*$-algebras, has a Kolmogorov decomposition if and only if it is completely positive; that is, the restriction of the kernel to any finite set of index points gives a completely positive map. The result may be viewed as a generalisation of the Stinespring dilation theorem from single point to multi-point index sets. This talk presents the analogue of the Haagerup-Paulsen-Wittstock decomposition theorem for $L(A,B)$-valued kernels (where now $B$ is assumed to be injective). It happens that in general complete boundedness of the kernel (ie, complete boundedness of the map resulting from restriction of the kernel to any finite index set) is not quite enough to ensure a decomposition: a certain regularity condition must also hold. This condition can be seen to be automatic if the kernel is completely positive or the index set is countable. This is joint work with Tirthankar Bhattacharyya and Chris Todd.