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Unique state extension and hereditary $$C^*$$-algebras. (English) Zbl 0694.46039
Let A be a $$C^*$$-algebra and let B be a $$C^*$$-subalgebra of A. This paper provides some necessary and sufficient conditions for every state of B to uniquely extend to a state of A. In particular, it is shown that every state of B uniquely extends to a state of A if and only if B is a hereditary $$C^*$$-subalgebra of A. For a $$C^*$$-dynamical system (A,G,$$\alpha)$$, it is also shown that an $$\alpha$$-invariant $$C^*$$- subalgebra B of A is a hereditary $$C^*$$-subalgebra if and only if the $$C^*$$-crossed product $$B\times_{\alpha}G$$ is a hereditary $$C^*$$- subalgebra of the $$C^*$$-crossed product $$A\times_{\alpha}G$$.
Reviewer: Masaharu Kusuda

##### MSC:
 46L30 States of selfadjoint operator algebras 46L55 Noncommutative dynamical systems 46L05 General theory of $$C^*$$-algebras
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##### References:
 [1] Batty, C.J.K., Kusuda, M.: Weak expectations inC *-dynamical systems. J. Math. Soc. Japan40, 662-669 (1988) · Zbl 0672.46040 [2] Kusuda, M.: HereditaryC *-subalgebras ofC *-crossed products. Proc. Am. Math. Soc.102, 90-94 (1988) · Zbl 0653.46063 [3] Pedersen, G.K.:C *-algebras and their automorphism groups. London: Academic Press 1979 · Zbl 0416.46043
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