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I am interested mainly in ring theory and homological algebra. Now I want to know about the research methods of AW*-algebras. So I want to know the answer to the question:"what methods have been used to study AW*-algebras up to now?" Thank you very much!

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I suppose a complete correct answer to this question would be some positive integer. I doubt that is really what you want. –  Chris Godsil Jan 16 '13 at 19:37
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I think this approach "puts the cart before the horse". If you are interested in AW-star algebras, and presumably have seen the definition somewhere, then you should try to understand the proofs, see what background they are using, and work backwards from there. –  Yemon Choi Jan 16 '13 at 20:35
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Also, if you are mainly interested in ring theory and homological algebra, why are you looking at AW-star algebras? It isn't clear to me what you are looking for when you say "the message of the research methods of AW-star algebra". –  Yemon Choi Jan 16 '13 at 20:37
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@Tom: $AW^\ast$-algebras were Kaplansky's attempt to "algebraicize" the basic theory of von Neumann algebras, by imposing conditions on a noncommutative ring asserting that it has enough idempotents to do lots of nice things. IIRC they have certain homological finiteness properties, which is related to having lots of idempotents. Nevertheless I too feel that this question is starting with a vague wish -- "methods used to study what about $AW^\ast$-algebras?" would be my response, regardless of any issues with the English language –  Yemon Choi Jan 17 '13 at 0:58
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Perhaps the question could be interpreted as "what current research is being done on $AW^\ast$-algebras?" But even then, I feel that's a bit too vague, it isn't clear what is desired from an answer except what a search with Google or MathSciNet would give. –  Yemon Choi Jan 17 '13 at 1:01
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