Summary
We generalise the theory of infinitely divisible positive definite functions f:G→ℂ on a group G to a theory of infinite divisibility for completely positive mappings Φ: G→ℬ(ℋ) taking values in the algebra of bounded operators on some Hilbert space ℋ.
We prove a structure theorem for normalised infinitely divisible completely positive mappings Φ which shows that the mapping Φ, its Stinespring representation and its Stinespring isometry are of type S (in the sense of Guichardet [Gui]). Furthermore, we prove that a completely positive mapping is infinitely divisible if and only if it is the exponential (as defined in this paper) of a hermitian conditionally completely positive mapping.
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Fannes, M., Quaegebeur, J. Infinitely divisible completely positive mappings. Probab. Th. Rel. Fields 79, 369–403 (1988). https://doi.org/10.1007/BF00342232
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DOI: https://doi.org/10.1007/BF00342232