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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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