By Corneliu Constantinescu
V. 1. Banach areas -- v. 2. Banach algebras and compact operators -- v. three. common idea of C*-algebras -- v. four. Hilbert areas -- v. five. chosen issues
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Additional info for C*-Algebras Volume 4: Hilbert Spaces
By Pythagoras' Theorem, the map FQF'+E, is an isometry. 2 b)). Take x E F L L . There is a pair z=y+z. 2 a)) we deduce that z = 0 . 2 e)), and Hence F L L c F , F" = F . 8 ( 0 a complemented subspace. 9 ( 0 ) Let E be a pre-Hilbert space and A a subset of E . If E is complete or if A is finite, then -4'' is the closed vector subspace of E generated by A . Let F be the closed vector subspace of E generated by A . 2 a),b)), F cA~L. If E is complete, then F is complete as well. If A is finite, then F is finitedimensional and therefore complete.
Then A L = (0) , A L L = E , and x does not belong to the closed vector subspace of E generated by A . 10 ( 0 ) Let E be a Hilbert space and F a vector subspace of E . T h e n the following are equivalent: a) F = E . 7) and the assertion now follows. 11 ( 0 ) Let E be a Hilbert space and take u E L ( E ) . A closed vector subspace F of E is said to reduce u if F and F L are U znvariant. 12 ( 0 ) Let E be a Hilbert space, F a closed vector subspace of E , and take u E L ( E ) . 2 Orthogonal P~ojectzonsof Hzlbert Space a ) F is u-invariant iff b) F reduces u iff a) is easy to see.
C) Take X , Y E E I F and x 1 , x 2 X~ , y , , y , ~ Y . T h e n so that by a ) . This proves the existence of g . The uniqueness is trivial. d ) is easy to check. 10 The map is a n injective homomorphism of unital real algebras. Identifyzng G with its image wzth respect to the above map, M becomes a two-dimensional complex vector space and the map ((aP , , y , 6 ) , (a',P', r ' , 6 ' ) ) * ( a + Pi)(crt - P'i) + (Y + 6 i ) ( ~-' b'i) is a scalar product generating the euclidean n o r m o n M The proof is a straightforward verification.
C*-Algebras Volume 4: Hilbert Spaces by Corneliu Constantinescu