Abstract: We prove that a pure state on a $C^*$ algebra or a JB algebra is a unique extension of some pure state on a singly generated subalgebra if and only if its left kernel has a countable approximative unit. In particular, any pure state on a separable JB algebra is uniquely determined by some singly generated subalgebra. By contrast, only normal pure states on JBW algebras are determined by singly generated subalgebras, which provides a new characterization of normal pure states. As an application we contribute to the extension problem and strengthen the hitherto known results on independence of operator algebras arising in the quantum field theory.
Keywords: JB algebras, $C^*$ algebras, pure states, state space independence of Jordan algebras
Classification (MSC2000): 46L30, 17C65, 81P10
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