1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

16.4. Counterexamples 427

After making this reduction, we consider the operator T E Mn(A) de-

fined as follows:

0 bi 0 0 0

(^0 0) b2 0 0
0 0 0 0 0
T=
0 0 0 0 bn-1
bn^0 0 0 0
Step 2 is to convince yourself that it suffices to show C* (T) contains the
matrices
[

~ ~ ~I , [~ b~ ~I , · · ·, [H : : : ~I ,

0 0 0 0 0 0 0 0 · · · bn
together with all of the matrix units of Mn(C) C Mn(A). If this is the case,
then matrix unit manipulations show we can generate all of Mn(A).

Step 3 is to use functional calculus on TT* (remember we separated the

spectra of the bJ's) to show that C*(T) contains all the matrices

II

0

!I·[!

0

ol lo

0

b~I

(^0) b~ 0 0 0
0 0
b ' ' b
0
It follows that C* (T) contains all the matrices
1~
0

f I · If

0

ol lo

0

btl

0 b2^0 0 0

b ' ' b

o· 0 0 0

and

[bf

0

~r I~

0

0 b-1 ol lo 0... 0 I

~ ' ... ' ~ ~ ·.·.· ~ '

2

.....
0 0 0 0 0 b-n^1

since bi is the unique positive square root of bJ and C* -algebras are inverse
closed. · ·

. The final step is to play around with all of the matrices we now know
belong to C* (T) and to show that all of the matrix units must also be in