|
In the theory of von Neumann algebras, the Kaplansky density theorem, due to Irving Kaplansky, is a fundamental approximation theorem. The importance and ubiquity of this technical tool led Gert Pedersen to comment in one of his books〔Pg. 25; Pedersen, G. K., ''C *-algebras and their automorphism groups'', London Mathematical Society Monographs, ISBN 978-0125494502.〕 that, :''The density theorem is Kaplansky's great gift to mankind. It can be used every day, and twice on Sundays.'' ==Formal statement== Let ''K''- denote the strong-operator closure of a set ''K'' in ''B(H)'', the set of bounded operators on the Hilbert space ''H'', and let (''K'')1 denote the intersection of ''K'' with the unit ball of ''B(H)''. :Kaplansky density theorem.〔Theorem 5.3.5; Richard Kadison, ''Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory'', American Mathematical Society. ISBN 978-0821808191.〕 If is a self-adjoint algebra of operators in , then each element in the unit ball of the strong-operator closure of is in the strong-operator closure of the unit ball of . In other words, . If is a self-adjoint operator in , then is in the strong-operator closure of the set of self-adjoint operators in . The Kaplansky density theorem can be used to formulate some approximations with respect to the strong operator topology. 1) If ''h'' is a positive operator in (''A''-)1, then ''h'' is in the strong-operator closure of the set of self-adjoint operators in (''A''+)1, where ''A''+ denotes the set of positive operators in ''A''. 2) If ''A'' is a C *-algebra acting on the Hilbert space ''H'' and ''u'' is a unitary operator in A-, then ''u'' is in the strong-operator closure of the set of unitary operators in ''A''. In the density theorem and 1) above, the results also hold if one considers a ball of radius ''r'' > ''0'', instead of the unit ball. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Kaplansky density theorem」の詳細全文を読む スポンサード リンク
|