|
In mathematics, the continuous functional calculus of operator theory and C *-algebra theory allows applications of continuous functions to normal elements of a C *-algebra. ==Theorem == Theorem. Let ''x'' be a normal element of a C *-algebra ''A'' with an identity element e; then there is a unique mapping π : ''f'' → ''f''(''x'') defined for ''f'' a continuous function on the spectrum Sp(''x'') of ''x'' such that π is a unit-preserving morphism of C *-algebras such that π(1) = e and π(ι) = ''x'', where ι denotes the function ''z'' → ''z'' on Sp(''x'').〔Theorem VII.1 p. 222 in Modern methods of mathematical physics, Vol. 1, Reed M., Simon B.〕 The proof of this fact is almost immediate from the Gelfand representation: it suffices to assume ''A'' is the C *-algebra of continuous functions on some compact space ''X'' and define : Uniqueness follows from application of the Stone-Weierstrass theorem. In particular, this implies that bounded normal operators on a Hilbert space have a continuous functional calculus. 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「Continuous functional calculus」の詳細全文を読む スポンサード リンク
|