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This book explores the algebraic and topological properties of non-commutative Lp-spaces, Arens algebras, trace algebras, Orlicz spaces, and log-integrable algebras, which play a key role in functional analysis and abstract algebra. It examines their connections to operator theory, measure theory, and non-commutative integration.Focusing on non-commutative integration theory, the book extends classical analysis to applications in quantum mechanics and mathematical physics. It is intended for master's students, doctoral candidates, and researchers, providing insights into Banach and operator algebras, as well as non-commutative integration.