194 W. Каир and H. Upmeier

/ ni\ 1 Ô

The Cayley transformation a = exp I ——E),E=-{e-e*) —-e g maps D biholo-

\ 2 / 2 oz

morphically onto the bounded balanced domain AczU (note that J=0 in this case). Since for every ae U the vector field E is tangent to U^ we get criDJ = A^ and

( 4 . 4 ) a(z)==i{z-a)iz-ha)-^

for all zeD and a = fe By the same argument we find for the symmetry s = exp(^(e-b.*)A)

for all zeD. Using (4.3) we obtain

i

z - w = -s"{a){z,w)

where z - w denotes the Jordan product in U and s"{a): U xU-^U is the second derivative of the symmetry 5 at the fixed point a.

( 4 . 5 ) Proposition. Let AczV be an associative subalgebra. Then there exists a compact space S and a triple isomorphism ц: U^ -^^{S) such that

( i ) p ( ß^ ) = {/6<^(S,R):/>0}, (ii)MßJ = {/e<^(S):Im(/)>0}, (in) fi{àJ = {fenS):\f\<l}.

Proof . By construction A is the bounded symmetric domain associated with the hermitian Jordan triple system (L/, *) and P^ is a flat subsystem of U. By [7, (4.7)] there exists a compact space S together with an isomorphism /x: U^-^^{S) of Jordan triple systems such that /u(l^) = ^(5, R) and lu(A^) is the open unit ball of ^(S). The function [л{е) takes only the values ± 1, so we may assume /i(e) = 1 e'^{S). Since <t{DJ = A A and бг(О^) = zl^ n F the result follows with (4.4). Q.e.d.

( 4 . 6 ) Proposition. Define R' = {xeV: x invertible] and Q- = {x^ :x€V]. Then (i) 0 = exp(F)=6nJR.

( ii ) For every xeQ the cone Q is invariant under P{x)eGL{U). (iii) Q=^Q={zz:zeU}. Q is the interior ofQ and also the connected e-component of the open set R<=:V.

Proof (i) follows from (4.5) applied to Л = I^^-i for every txeV, By [1, p. 317] we have

( 4 . 7 ) P(expa) = exp(2L(a))

for all aeF. 2L(a)---=|a-->^*^^N9 implies P(expa)D=D and hence (ii).

oz L dz ÔZJ

Now fix an element ae V. By (4.5) a may be considered as a function cce^ {S) = X. Therefore «ей if aeQ. Suppose on the other hand that aeO is not in Q. Then a