By P. de la Harpe

ISBN-10: 0387059849

ISBN-13: 9780387059846

ISBN-10: 3540059849

ISBN-13: 9783540059844

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Extra info for Classical Banach-Lie Algebras and Banach-Lie Groups of Operators in Hilbert Space

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Co) , then algebra ii) In other words, A if A sl(~ ; Co) is replaced is a derivation of is also a derivatic~ of the associative simple Co(H). Proposition 2A can alternatively be d e d ~ e d results, due to ~artindale associative from general [117], about Lie derivations rings (see his theorem 2 with R = Co(~) of certain and : Tin ( ~ 9 [ ) ) . iii) The people who live in are in fact continuous It is not necessarily o(~ 'JR; Lin) and (because they have adjoints; the case for those in 31 ~l(~; so(~ ,JQ; Lin) [45], 12.

IS~,q) p,q e N * with takes the same It follGvs that the p ~ q , value, it say exp(iw) two last exp~essions can be written as @su(Eq,_p - Ep,_q) ~su(E_p,q- = exp(-iw) E_q,p) : ex~(iw) for Let finally exp(iw) T (fn)n ~ N* (f-n)n ~ N* " (F_p,q - F_q,p) p,q ~ N* be the operator in the span of in the span of all (Fq _p - Fp,_q) Then = ~TSU " • 56 with on'which p ~ q . 35 Proposition 10B. Let *-automorphism of V e 0(~ IX e U ( ~ ) 2) = ~ ~ = . @=@V Otherwise {1} . there exists I XJ~ = J2Xl ~-~ Let @ be a an operator such that VXV* @ , up to a sign.

This b~A; subalgebras algebra imposed to the tl~ consideration (as in Schue [I 53] 7B(type is that a given t o g e t h e r with a type c~ on the the following be a regular then there contains is . involution 2 and 6 seem very restrictive. a general : is isomorphic The only point to stress a Lie The conditions Conjecture WA , of it. in definition point f o r h , say . guess by now what propositions two) and 7C are. sub~lgebra WA in ~ the inspection n e e d e d for li). • The r e a d e r will easily one), N to from proposition of subsys terns of root s a n d of ad hoc section 3) of roots conjecture results due to Kaplansky, associative algebras semi-simple known dense 46 complex complex L*-algebra subalgebra.

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Classical Banach-Lie Algebras and Banach-Lie Groups of Operators in Hilbert Space by P. de la Harpe


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