Abstract
Inspired by Rota’s Fubini Lectures, we present the MV-algebraic extensions of various results in probability theory, first proved for boolean algebras by De Finetti, Kolmogorov, Carathéodory, Loomis, Sikorski and others. MV-algebras stand to Łukasiewicz infinite-valued logic as boolean algebras stand to boolean logic. Using Elliott’s classification, the correspondence between countable boolean algebras and commutative AF C*-algebras extends to a correspondence between countable MV-algebras and AF C*-algebras whose Murray-von Neumann order of projections is a lattice. In this way, (faithful, invariant) MV-algebraic states are identified with (faithful, invariant) tracial states of their corresponding AF C*-algebras. Faithful invariant states exist in all finitely presented MV-algebras. At the other extreme, working in the context of σ-complete MV-algebras we present a generalization of Carathéodory boolean algebraic probability theory.
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Mundici, D. (2009). Rota, Probability, Algebra and Logic. In: Damiani, E., D’Antona, O., Marra, V., Palombi, F. (eds) From Combinatorics to Philosophy. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-88753-1_9
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DOI: https://doi.org/10.1007/978-0-387-88753-1_9
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