During this e-book we describe the user-friendly thought of operator algebras and elements of the complicated idea that are of relevance, or most likely of relevance, to mathematical physics. accordingly we describe quite a few functions to quantum statistical mechanics. on the outset of this undertaking we meant to hide this fabric in a single quantity yet during advance ment it used to be learned that this could entail the omission ofvarious fascinating themes or info. as a result the publication was once break up into volumes, the 1st dedicated to the final idea of operator algebras and the second one to the purposes. This splitting into concept and purposes is traditional yet a little arbitrary. within the final 15-20 years mathematical physicists have discovered the significance of operator algebras and their states and automorphisms for difficulties of box idea and statistical mechanics. however the thought of two decades aga was once principally built for the research of team representations and it was once insufficient for lots of actual functions. therefore after a quick honey moon interval during which the hot chanced on instruments of the extant concept have been utilized to the main amenable difficulties an extended and extra attention-grabbing interval ensued within which mathematical physicists have been compelled to redevelop the idea in appropriate instructions. New suggestions have been brought, e. g. asymptotic abelian ness and KMS states, new concepts utilized, e. g. the Choquet thought of barycentric decomposition for states, and new structural effects received, e. g. the lifestyles of a continuum of nonisomorphic type-three elements.
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