1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

160 4. Constructions

and

w i,3 ( >-.j (b)) ( (i ® (2 ® ... ® ei ® ... ) = Vi73 ( (bej) ® (i ® (2 ® · · · ® ei ® · · · )

= ei ® (bej) ® (i ® (2 ® ... ® ei ® · · ·

= 1/Jj (b) ( (i ® (2 ® .. · ® ei ® · · · ).

We next check, for (k E 'Hfk with i -/= ii -/= · · · -/= im, that

Wi,3(>-.i(a))(ei ® (i ® · · · ® (m ® ei ® · · ·)

and if m = 1,

= Vi73 ( aei ® (i ® ... ® (m ® ei ® ... )

= aei ® (i ® ... ® (m ® ei ® ...

= 1/Ji(a)(ei ® (i ® ... ® (m ® ei ® ... )

Wi,3(>-.j(b))(ei ® (i ® · · · ® (m ® ei ® · · ·)

= Vi73 ( ((b(i - ej\ej, b(i)) + e\ej, b(i)) ® ei ® · · ·)

= ei ® (b(i - ej\ej, b(i)) ® ei ® · · ·

= u((b(i) ® ... ® (m ® ei ® ... )

= 1/Jj(b)(ei ® (i ® ... ® (m ® ei ® ... ),

while if m > 1,

wi,3(>-.j(b))(ei ® (i ® · · · ® (m ® ei ® · · ·)
= Vi73( ((b(i - ej\ej, b(i)) ® (2 + (ej, b(1)(2) ® ... ® (m ® ei ® ... )
= (ei ® (b(i - ej\ej, b(i)) ® (2 + \ej, b(i)(2) ® · · · ® (m ® ei ® · · ·
= u((b(i) ® ... ® (m ® ei ® ... )
= 1/Jj(b)(ei ® (i ® ... ® (m ® ei ® ... ).
Hence, wi,3(>-.i(a)) = 1/Ji(a) for every a E Af and wi,3(>-.j(b)) = 1/Jj(b) for
every b E Aj.
Claim. Let <Pi: JIB(H) -+ JIB('H(j)) be the compression. Then, <Pi is "multi-
plicative on reduced words," i.e., for every ak E Aik with ii -/= · · · -/=in, one
has
'l>i(Ai 1 (ai) · · · Ain(an)) = 'l>i(Ai1 (ai)) · · · 'l>i()..in (an)).
The proof is by induction. Let ak E Aik with ii -/= · · · -/= in+l be given. If
in+l -/= j, then

Ain+l (an+i)P?-l(j) = P?-l(j))..in+l (an+i)P?-l(j)

and we are done. Now, suppose that in+l = j. Then, we have

Ain+i ( an+i)P?-l(j) = (P~D + P?-l(j) )Ain+i (an+i)P?-l(j).