50

M. L. RACINE

T 1+ M^ C M^

- v ( d )

.0 n (d3(^Q n ? ))"

Multiplying out and using the fact that M' C © while r 0, we obtain

the desired result s in both c a s e s .

LEMMA 17. Let H

q. e. d.

C l d 2

H.

H.

d . € &Q ; v(

C ] L

) v ( c

3

) ; v f d ^ , v(d

2

) v f c ^ ; v ( d

3

) , v(d

4

) v(c

3

'). Le t

M„ = {H^AI A € tt(&Af' $„), H i ? © . } . If we make the obvious identifications of

4 4 ' 4 0 4 4

M and M , determined by H and H as in Lemma 15, with subalgebras

of M , . then

4

^ r pr p ©

pr pr © ©

M'4

= I f f f f

r r r r

? ? ? ?

+ M' + M' where r = v(c ) - v(c ).

PROOF. It is eas y to se e that e + e (= unit element of M ) and

e ^ + e ^ (= unit element of M2)'

0

are contained in M„. Also a

r

„„-,' a

r[14

, „..,

3 3 4 4 4 [13] V

-1 -1 ~ v ( c i )

a, -,, a, , e M if and only if a g (c , c ) © = c © = ^p . We wish

to show that ©, „ c M' Let x

e

©. Since v(d, ), v(d

0

) v(c, ).

13 4 1 2 1 ' •