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Summary: Contemporary Mathematics
Four Lectures on Noncommutative Dynamics
William Arveson
Abstract. These lectures concern basic aspects of the theory of semigroups of
endomorphisms of type I factors that relate to causal dynamics, dilation the-
ory, and the problem of classifying E0-semigroups up to cocycle conjugacy. We
give only a few proofs here; full details can be found in the author's upcoming
monograph Noncommutative Dynamics and E-semigroups, to be published in
the Springer series Monographs in Mathematics.
Contents
1. Dynamical Origins: Histories and Interactions 1
2. Generators of Dynamics and Dilation Theory 13
3. The Role of Product Systems. 23
4. Spectrum of an E0-Semigroup. 45
References 54
1. Dynamical Origins: Histories and Interactions
The dynamics of one-parameter groups of automorphisms of B(H) has been
completely understood since the 1930s, following work of Wigner, Stone, and the
multiplicity theory of Hahn and Hellinger. After reviewing these basic issues, we
show that if one takes into account a natural notion of causality for such dynamical
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