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Summary: GENERATORS OF NONCOMMUTATIVE DYNAMICS
WILLIAM ARVESON
Abstract. For a fixed C
-algebra A, we consider all noncommutative
dynamical systems that can be generated by A. More precisely, an A-
dynamical system is a triple (i, B, ) where is a -endomorphism of a
C
-algebra B, and i : A B is the inclusion of A as a C
-subalgebra
with the property that B is generated by A (A) 2
(A) · · · . There
is a natural hierarchy in the class of A-dynamical systems, and there
is a universal one that dominates all others, denoted (i, PA, ). We
establish certain properties of (i, PA, ) and give applications to some
concrete issues of noncommutative dynamics.
For example, we show that every contractive completely positive lin-
ear map : A A gives rise to to a unique A-dynamical system (i, B, )
that is "minimal" with respect to , and we show that its C
-algebra B
can be embedded in the multiplier algebra of A K.
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