By P. de la Harpe
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Extra info for Classical Banach-Lie Algebras and Banach-Lie Groups of Operators in Hilbert Space
Of ~ is m a x i m a l all its elements it follows which over Is among the are semi-simple. that Caftan subalgebras 4 are Cartan sub algebras of ~ of in the sense. Lie algebra sense. Let h let and let Then invariant h ~ h be a finite be a Caftan there exists a c-involution 11, & dimensional be a finite (Kostant th~or~mes subalgebra ),there is invariant. din~nsional subslgebra (, expos~ be a Caftan standard which of ~ to definition Conversely, let c~ this d e f i n i t i o n th~ n~tter of this chapter a Caftan subalgebra sub al gebras ~ usual so that be a semi-simple the abelian In particular, When to chapter if A critic of semi-simplicity subalgebra among abelian ideals.
It . do~n to the f o l l o w i n g ~ of . and let for all Let . ~ rl = ~-l r2 w lJ. g *-automorphi~m of for These remarks b o i l Prooosition of f r o m s_~ 1 to ~ s2 r2(-W(Y)* ) = W(Y) set of all inv~lutive set of *-isomorphism real forms let the is onto. *-automorphism *-isomorphisn Then is a map from b e an involutive Conversely, which ~ g, and this map Let n o w clearly of there be a complex corresporhence, conjugacy proposition. involutive as above, classes ~ Lie algebra. Then bet~/een involutive ~-automorphisms of ~ .
Let finite dimensic~al (3) A(X) Moreover, properties, operator finite for all the restriction that 3. of A dimensional E A(X) of~ ; Co) s in on s is a l i n e a r map derivation, = [D,X] for all and namely X ~ s. of the first  §6 exercice la or of sl(~ ; Co). F(r every of dimensi on at least 2, there s u o h that X ~ sl(E). YE be a derivation to E is uniquely a scalar multiple of operator YE e o ( ~ 28 defined by these cf the identity. _ o ( ~ 'Jg9; Co)" J&q-invariant subspace least 5, there exists an of term~ • up to the addition of B.