Abstract: In the late eighties, Powers initiated the study of 1-parameter semigroup of endomorphisms on B(H). This was further studied intensively by many others during the last three decades led by William Arveson. Although from the physics point of view, 1-parameter theory is the most important one, from the mathematical perspective it is not necessary to restrict oneself to 1-parameter, i.e. the half-line [0, ?) and we could replace the half-line by any reasonable semigroup like convex cones in higher dimensional Euclidean space. One of the nice features is that the basic theory stays intact while there are significant differences between the 1-parameter theory and the n-parameter theory. I will explain one such phenomenon. In particular, I will define the basic examples of E0-semi-groups, i.e. the CCR and CAR flows associated to isometric representations of the semigroup P. In the multi-parameter world, CCR flows need not be isomorphic to its opposite, a sharp contrast to the one parameter situation.
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