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Summary: ON THE INDEX AND DILATIONS OF
COMPLETELY POSITIVE SEMIGROUPS
William Arveson
Department of Mathematics
University of California
Berkeley CA 94720, USA
28 August 1996
Abstract. It is known that every semigroup of normal completely positive maps
P = {Pt : t 0} of B(H), satisfying Pt(1) = 1 for every t 0, has a minimal
dilation to an E0-semigroup acting on B(K) for some Hilbert space K H. The
minimal dilation of P is unique up to conjugacy. In a previous paper a numerical
index was introduced for semigroups of completely positive maps and it was shown
that the index of P agrees with the index of its minimal dilation to an E0-semigroup.
However, no examples were discussed, and no computations were made.
In this paper we calculate the index of a unital completely positive semigroup
whose generator is a bounded operator
L : B(H) B(H)
in terms of natrual structures associated with the generator. This includes all unital
CP semigroups acting on matrix algebras. We also show that the minimal dilation
of the semigroup P = {exp tL : t 0} to an E0-semigroup is is cocycle conjugate to
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