# Jordan operator algebras by H.Hanche- Olsen, E. Stormer

By H.Hanche- Olsen, E. Stormer

Hanche-Olsen H., Stormer E. Jordan operator algebras (Pitman complicated Pub. application, 1984, 5500)(ISBN 0273086197)

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Ii) Denote by Im the imaginary part of relative to the real form h of hC , and similarly for 1 . This imaginary part is a coset, and we take the representative with the smallest magnitude relative to a certain norm. A/1 / I kIm. 2. A/ action. Here AM;1 is the group of real points of a torus associated to M. A/; t/. ƒ/ W x 7! A//ƒ . M/ be another parabolic subgroup. ƒ/ ! ƒ/: The Eisenstein series theory provides a meromorphic continuation of this operator to all ƒ 2 aM;C . The following proposition serves to define the discrete part of the trace formula.

N; F/. Suppose s D 1. The characteristic polynomial defines a continuous map G ! F n . 1/ is the set of unipotent elements in G. g; f /. sui ; f /. 7. s/0 , in which s is central. s/0 is equal to that of f on G in a neighborhood of su in Tu, as follows. f /. s/0 nG is locally compact. s/0 , equals the characteristic function of C in MnG. 6). g; fs /. sui / is the decomposition into orbits in M of the set of elements with semisimple part conjugate in M to s. 7. sui ; fs /: iD1 A unipotent element u has conjugates arbitrarily close to the identity.

The proof consists of two parts. 11. G0 /, there is f , such that . 1. PROOF. G0 / generated by the equivalence classes of irreducible tempered G0 -modules 0 by ˆ. 0 / D . f / if 0 is square-integrable, and it corresponds to , and by ˆ. 0 / D 0 if 0 is irreducible, tempered but not square-integrable. It is clear that ˆ is a good form in the terminology of [BDK86] or [F95], hence a trace form by the Theorem of [BDK86] or [F95]. Namely, there exists f 0 on G with ˆ. G0 /. G0 /. G0 /, there is f on G) is analogous.