# Asymptotic quantization : based on 1984 Naples lectures by Abhay Ashtekar

By Abhay Ashtekar

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Is a c o n t i n u o u s that the same map with is true norm for e-isomor- phism. 5. Proposition. Hausdorff then space and Let let To p r o v e it is s u f f i c i e n t that to show : A ~ ~ : Ts(f) = Tf(s) is a l s o assume then and an that let from first ker T in A to S A be a compact into A@ we that show ker T in S the C(S), so w i t h o u t loss of g e n e r a l i t y {le} an that if A does by p u t t i n g that has T functional we can not p o s s e s s a unit so w i t h o u t loss of g e n e - a unit.

S are (s) ' From = X'-- U (% weak o , converges ~ of v s and convergent to s £ S' O f we " have (31) 42 Hence 2 I= can 116x where 1111 > I - 6E o. o represented in t h e f o r m be 2 = 12~ x + Av 2 where On the other hand the measure z ~ 2 ( { X o }) = 0 . k o-- 1111 > ~(1111 Notice now 1-6e o k 6~o> + var(~ 2 )) . then + var(~2)) that, (Iv s~ i ( U ) =M(ll1[ I since ~< M< + I 1121 + v a r ( A ~ 2 ) ) then for any (32) . 11,126 ~ we have Ix1+x21-[x11>_M(Ix I+12[- Ix11- Ix21). Adding by sides (32) and (33) (33) we g e t IX I + X21 > _ M ( I X I + 121 + v a r l A ~ 2 ) ) >_M v a r (~0) _> M lIT*~ s II • o Let The now ~ point be x° any being regular in t h e ~ IX1 + X21, h e n c e b y t h e measure Choquet foregoing on SA boundary which represents of A we and let have T*~ s l~({Xo}) o I~ inequality ~I{Xo}}i >_MIIT*~ s II o and this proves (i) and s Fix ~ s now By that the our s s o a compact £ S x o subset K of ChA s 6 Sx on S , x 6 K 6 S.

To e n d the p r o o f let g be any e l e m e n t A of with II gll ! I. 6. A onto Remark. IITII ! I+ e. Let a function T algebra II T - I ( T f ' T g ) Hence, be a l i n e a r f'gll by P r o p o s i t i o n B isomorphism such algebra that ! 5 we h a v e from a function f,g6 for IITII ! I + e A. and so we get I I T f ' T g - T ( f ' g ) ll ~ l l T I l ' l l T - 1 1 T f ' T g ) - f ' g l l ! ~ ( I + e)'llfll "l[gll for f , g 6 A. The above consideration proves that, for any function the class of isomorphisms defined by D e f i n i t i o n bigger § 6.