By Corneliu Constantinescu

V. 1. Banach areas -- v. 2. Banach algebras and compact operators -- v. three. normal thought of C*-algebras -- v. four. Hilbert areas -- v. five. chosen subject matters

**Read or Download C*-Algebras Volume 4: Hilbert Spaces PDF**

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**Extra info for C*-Algebras Volume 4: Hilbert Spaces**

**Sample text**

C + d . 5 a + e , d @ e . 1, = (ACXIX). I I ~ F X =~ ~( ~~ F x ~ x )llrcxll2 , d =+ a . Take x E F . 1, 1 1 ~ 1 =1 ~ l l ~ ~ xIl IlI ~~ G x I I ~ = 1 1 ~ 1 -1 ~ 115 - ~cx1I25 1 1 ~ 1 1 ~ Thus and FCC. a & b & c & e + f . e. KG - K F = K$ - - AGKF - AFKG + A; = AG - KF is a projection in E . Take x E E . 1 a 3 F ~ X , -AFXEFI b). 2 a)).

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. 1 and put O E := ~ LEI Let ( E L ) r EbeI a family of pre-Hilbert spaces I I X E L E~ I a) @EL is a vector subspace of LEI , ~L El I l E 1 xLl12

N+m + a & d and d + e are trivial. + d . 1 Pre-Hilbert Spaces By b + c , x and eZeyare linearly dependent. Hence x and y are lineraly dependent. Remark. a ) It is easy to see that d) does not irnply b) (take y b) In e l , := - x ) . and so a ) does not imply c) in the case of arbitrary Banach spaces. 14 5. 1 (0) Let ( T , I , p ) be a measure space. The m a p is a scalar product. 2 (0) Let T be a set. 3 The m a p is a scalar product. e2 endowed with this scalar product is a Hilbert space (it is called the Hilbert space of square summable sequences).