Faithfulness refers to a property of states in a von Neumann algebra that ensures no non-zero positive element is annihilated by the state. A faithful state maintains that if the expectation of an element is zero, then the element itself must be the zero element. This concept connects deeply to normal states, which can be thought of as continuous linear functionals on a von Neumann algebra, and is essential when discussing Type III factors, where faithful states play a critical role in the structure and behavior of these algebras.
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