The automorphism group of the Toeplitz algebra generated by the Toeplitz operators, whose symbols are continuous functions on the circle beside finite...

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SCIENCE

IN C H I N A (Series A)

October2001

Automorphisms of the Toeplitz algebra with piecewise continuous symbol YAN Congquan

()~.]E),~..~)I,2,

SUN Shunhua (.~,]'~l~,~-P-)2

& DING Xuanhao ( 3 1. Laboratory of Mathematics for Nonlinear Sciences, Fudan University, Shanghai 200433, China; 2. Department of Mathematics, Sichuan University, Chengdu 610064, China; 3. Department of Basic Courses, Institute of Electronic Technology, Guilin 541004, China Correspondence should be addressed to Yen Congquan Received February 28, 2001

The automorphism group of the Toeplitz algebra generated by the Toeplitz operators, whose symbols are continuousfunctions on the circle beside finitely fixed points, is characterized.

Abstract

Keywords:

automorphism, ToepHtzalgebra, piecewise continuous symbol, Hardy space.

Let T be the unit circle, L 2 and H a be the square-integrable measurable function space and Hardy space respectively. For f i n L | ( Y ) , let Tf denote Toeplitz operator P M f l H 2, where P : L 2 --~ H 2 is the orthogonal projection. For a subalgebra A of L | , let J ( A ) denote the C * -algebra generated by all Tf, f E A. This paper aims at characterization for automorphism group Aut( J ( A ) ) of J ( A ) , where A is the continuous functions on all ~ but finitely fixed points. Let z l , z2, " " , z, be fixed points in ~Z, which are distributed anticlockwise. Let PC;,,] denote the functions, which are continuous on all ~ but z l , z2, "'", z, and are left continuous,have both right and left limits at every point. PCE, 1 is a subalgebra of L | ( T ) . If n = 1 and zl = 1, PC Ea ] is denoted specially by PC [1]. For the C * -algebra J ( C ( ' ~ ) ) generated by the Toeplitz operators with continuous symbols, because its commutator ideal (also the semi-commutator ideal) is the compact operator ideal on H 2 , every operator in J ( C ( T . ) ) has the form T~ + C, C E ~ . It is well known (see ref. [ 1 ] ) that for any a E Aut( J ( C ( T ) ) ) , there is a unitary operator U and an orientation preserving homeomorphism a on ~, such that a ( T ~ ) = U* T~U = T~o, + C. Conversely, for tiny orientation preserving homeomophism a on T, there exists unitary operator U satisfying U * Tc,U = T§ + C for any ~ in C ( ~ ) , hence a defined by ct (TO,) = U * T§ is in Aut( J ( C ( T ) ) ). According to the same construction: Tg + C of the Toeplitz operator algebra with continuous symbol in the odd sphere S 2~-1 , Yen I21 obtained the same results in S 2~-1 . Some results relative to this problem can be found in refs. [ 3 - - 5 ] . Since the commutator ideal of J ( P C [ , 3) is ~d', but the semi-commutator ideal is not 2~, the operator in J( PC[ ,] ) does not have the simple form as T§ + C. This induces some difficulties to characterize Aut( J ( PC[,,] ) ) . In a more complex case, Muhly and Xia [6] studied the automorphisms of J ( L * * ) and proved that if tr is an orientation preserving differentiable homeomorphism whose derivative satisfies d-type Lipschitz condition, then there is an automorphism on J ( L = ) ,

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AUTOMORPHISMS OF TOEPLITZ ALGEBRA

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such that a (T~) = T~~ + C for any ~ in L | (~'). The 8-type Lipschitz condition controls in fact the rotational rate of a. During the characterization of the Ant( J( PC[ n? ) ) in what follows, the action of this condition will be realized more clearly. At first, we give some lemmas. L e t X = ( ~ - t l t ) U [ 0 , 1 ] , and f o r 4 i n P C i n 1, denote

{4(x),

4 #(x) =

x E T-

x4++ ( l - x ) 4 _ ,

{1)x,

E [0,1],

where 4+ = lim 4(e'~), 4_ = lim 4 ( e " ) . t~0 +

t ~0-

I~lllllla 1 [7] . For 4 and ~b in PCII 1 , T~Tr - T~T~ E ~ , T~ is Fredholm if and only if 4 # does not vanish and index of T~ is minus the winding number of 4 ~ with respect to the origin. By ref. [ 8 ] p. 203, the maximal ideal J ( PCIll ) / " ~ is just X. The topology on X is defined as follows: i t s n e i g h b o u r h o o d s a r e ( x - e, x + e) C S -

{1t forx E ~ ' - {1t; ( x -

r

x + e) C ( 0 , 1 ) f o r x E ( 0 , 1); (1, e ~) I,J ( 1 - e , 1 ] f o r l a n d ( e - ~ , 0) I,J [ 0 , E ) f o r 0 , where e > 0. For any T~ E J ( PC[~I ), the Gelfand transform of ~r( T§ ) is ~ ( ~r( T~ ) ) ( x ) = 4 # ( x ) f o r x E X. Then we have Lemmaa 2. If i : 9ff-~ J ( PC I~~) is the embedding map, ~ ( T~ ) = 4 # , then the exact sequence holds as follows: 0 --~ ~K Define at : ~" ~

i

9 J(PC[, 1)

, ' C ( X ) --~0.

X by e(20_r

ao(e ia) =

~

,--< 2

8

1

rr

2

37r 0<--, 2 ~r

-- + - - , 0 <~ 0 <~ - - , ~r 2 2 0 3 3rr

2 ~< 0 < 2*r.

It is obvious that a0 is a horneomorphism by the definition of the topology on X. So we have the exact sequence by Lemma 2:

0~ ~

i , j(pcE13) ~

C('Z) ~ O,

where ~ r ~ ( T ~ ) ~ ao = 4# ~ do. Lenuna 3. The algebra J ( P C : ~ 1) is sin~y generated. Proof. Taking ~ in PC[1] so that 4 + - 4_ = 1 and the winding number of 4 # with respect to the origin is one, 4 # ~ a0 is a continuous function on 7~ with orientation preserving and the winding number is one. Then C('~) is generated by 4 # ~ at and 1. Because T~ Tg - T~T~ is non-zero compact operator, the C"-algebra generated by T~ and I contains 9g. Let B denote this algebra. For any M E J( PC[13 ), ~" ( M ) in C('~) belongs to the algebra generated by ~ ~ ( T~ ) and 1. So 7r ( M ) belongs to the algebra generated by Ir (T~) and I. This implies that there is such an N in B that l r ( M ) = r r ( N ) . Hence M - N E . ~ a n d M E B. Therefore J ( PCII2 ) = B. This completes the proof. Theorem 1. For a in Ant J ( PCEI 3), there exists a unitary operator U and an orientation preserving homeomorphism a~ on ~ such that a( M) = U" MU for any M E J( PC[I]) , and

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SCIENCE IN CHINA (Series A)

~* ( a ( T ~ ) )

Vol. 44

= $* (T~) o a~ for ~ E PCE1~9 Conversely, for any orientation preserving homeo-

morphism a " , there exists a unitary U satisfying ~ ( U ~ T~U) = ~ ~ (T~) ~ a ~ for any ~ in

PC[II, and a defined by a( M) = U ~ MU is an automorphism on J( PCN]). Proof. Suppose a E A u t ( J ( P C [ I ] ) ) . Since ~ ' i s the commutator ideal of J ( P C N ] ) , a (.Td') = ~ . By ref. [ 1 ] , there is a unitary U such that a ( M ) = U" MU for M in J( PCEI 3). For M in J(PCEI]), let ~ ( T r ( M ) ) = ~ r ( a ( M ) ) . Then ci is an automorphism on J ( PC ~1] ) / • . a associated with the Gelfand transform ~: J ( PC ~t ] )/0~'--~ C ( X ) induces an automorphism ci " on C ( X ) such that ci " ( ~ # ) = ~ ( 7r ( a ( Tg ) ) ) for any ~ in PC[ll. Since A u t ( C ( X ) ) ----_ H o m e o ( X ) , which stands for the auto-homeomorphism group on X, there is a~ E H o m e o ( X ) , such that ci " ( f ) = f * a~ f o r f in C ( X ) .

Therefore ci " (~ # ) = ~ # ~ a, =

~(Ir(a(T+))). Leta2 a0, i . e

9

= a~ 1~ a~* ao, thena~' E H o m e o ( 7 ) a n d ~ # * ao* a2

~ * (ct(T~)) = 4 #

~

a0

o

a~*

~

~:

x~

(T~)

o

a~-x-

= ~(~r(a(T+)))

~

,

Next we show that a2 is orientation preserving. Taking ~ in PC[I] so that the winding number of ~ # with respect to origin is one, we have indT~ = -

1. S i n c e a ( T + ) = U ~ T+U, theninda(T+)

=-

1. Owing t o r e f . [ 6 ] , it is known

that ind T~ and inda ( T+ ) are minus the winding number of ~" ( a ( T~ ) ) and ~ ~ ( T+ ) respectively. Therefore a~ is orientation preserving by ~ ( a ( T §

= ~" (T§

* a~ .

On the other hand, suppose a ~ be an orientation preserving homeomorphism on T. Define ~ " : J ( P C E 1 ]) --~ C(2") by ~ " ( M ) = ~(M) * a ~ Using the statement in ref. [ 9 ] ,

( J ( PC:~] ), ~ " )

is an extension of ~

by C ( " 2 ) ,

and the extension is equivalent to

( J ( P C I ~ ] ) , ~ " ) . Hence there is a unitary U such that U ~ ( J ( P C [ 1 ] ) ) U = J(PC[~?) and ~"(M) = $'(U~MU) for any M ~ J(PCI1]). For any ~ ~ PC~I;, we have that ~" ( U" T~U ) = ~ " ( T~ ) = ~" ( T§ ) * a ~ , which implies that a defined by a( T§ ) = U" TgU is an automorphism on J ( P C [ I ~ ) . This completes the proof. It is easy to see that a " satisfying ~" ( a ( T~ ) ) = ~ ~ ( T~ ) * a ~ is uniquely determined by a in A u t ( J ( P C [ 1 ] ) ) ,

and U with a ( T ~ )

= U" T~U is uniquely determined by a up to a uni-

modular constant. For any orientation preserving homeomorphism a ~ , unitary operator U satisfying ~" ( U ~ T~U) = ~ ~ ( T~ ) o a ~ is not unique. If U1 and U2 satisfy this condition, then commutator [ U~ U2 * , T~ ] is compact. Therefore, by Theorem 1, we can give an exact sequence similar to that given in ref. [2 ] : O--~T--~ {U ~ U ( H ~) I [ U , T~] ~ 3g, V ~ ~ PC[Ilt--~ Aut(J(PC[~])) --~ Home.+ (~,~) --~ 0, where U( H 2) is the unitary operator group on H 2 and Home. § ('2) is the orientation preserving homeomorphisms on "2. Theorem 1 characterizes Aut( J( PCtt] ) ) in terms of Home.+ (3"). However, sometimes we prefer to use Home. ( X ) rather than Home.§ ( ' 2 ) . To do this, we need only denote H o m e o + ( X ) = t a ~ H o m e o ( X ) I a~ 1 ~ a

~ ao ~

Home.+

( ' 2 ) t.

Then for a

A u t ( J ( P C [ ~ I ) ) , there is a unitary U and a in Home.+ ( X ) such that ~( U" T§ O'.

= ~(T§

~

No. 10

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From now on we will investigate an interesting problem: for a in Aut( J(C(7) ) ), it is well known that there is a o in Homeo+ (T) such that a ( T ~ ) = T~~ + C, C E ~ . Is there a similar result for a in Aut( J( PC[I] ) ) ? By Theorem 1, this problem should be stated actually as follows : for which a in Homeo+ ( X ) , does a in Aut( J( PC[l] ) ) associated with a satisfy a (T~) = T§ + C for any ~ in PCE1] , where C E 2 U a n d al is a homeomorphism on T? Theorem 2. For a E Homeo+ ( X ) , there exists a in Aut(J(PC:l])) and al in Homeo(T) satisfying ~ ( a ( T ~ ) ) = ~(T~) o a and a ( T ~ ) = T~o~ + Cfor any ~ in PC[1], if and only i f a ( x ) = x for a n y x i n [ 0 , 1 ] . Proof. For a in Homeo§ ( X ) , there is a in Aut(J(PC[ll)) with ~ ( a ( T ~ ) ) = ~(T~) o a by Theorem 1. Suppose that there is ax E Homeo(T) with a ( T ~ ) = T~o, + C. Since T~~ E J ( PCEl] ) for any ~ in PC,.II, we have a I ( 1 ) = 1. Combining ~ ( a (T§ ) = ~(T~) o a = ~# ~ a a n d ~ ( a ( T ~ ) ) a for any 4 E PC[1].

= ~(T~o~) -- ( 4 ~ a l ) # , w e h a v e ( 4

o al) # = 4 # ~

Because there is a unitary U such that a(T~) = U ~ T~U = T~.~ + C for any 4 in P C t , 1 , we have - 1 = indU ~ T~U = indT~o, = indT~ . Hence al E Homeo+ ( ~ ) , which implies that ( ~ o 01)+ = 4 + a n d ( ~

o 01)_ = ~ - . So, for a n y ~ E PC[l~, we have

(4

o Ol)#(x)

f4

l

o 0"1(.,~) ' X ~

T -- t l t

,

x E Io,1].

On the other hand, we have 4 # ~ a(x) It is easy to show t h a t a ( [ O , 1 ] )

~4(a(x)),a(x) = [~#(a(x)),a(x)

E ? - {1}, E [0,1].

= [ 0 , 1 ] . In fact, if not, then there i s x o E

[0,1]with

a (Xo) E ~ - {1 t. Hence 4 # (Xo) = ~ ( a (Xo) ) for any 4 in PC[t]. It is easy to construct such a function ~ E PCtl~ that ~# (Xo) = Xo~+ + (1 - xo)~_ # ~ ( a ( X o ) ) . This contradiction implies our claim. Sincea(x) E [O,l]ifandonlyifx E [ 0 , 1 ] , then r ( x ) = 4'# ( a ( x ) ) for x in [0, 1], i . e . x~++ (1 - x ) 4 _ = a ( x ) ~ + + (1 - a ( x ) ) 4 _ . Hence ( x - a ( x ) ) 4 + = (x a ( x ) ) 4_ for any $ in PC[~, which implies that a ( x ) = x for x in [ 0 , 1 ]. Conversely, suppose a E Homeo+ ( X ) with a ( x ) = x for x E [ 0 , 1 ] . It is obvious that a ( 7 7 - {1}) = 77- 11}. Set Ol(X) = a ( x ) f o r x E ~ ' - {1} and a l ( 1 ) = 1, then Ol Homeo+ (T) a n d 4 # o a ( x ) = ( 4 ~ a l ) # ( x ) for any 4 ~ PC[~]andx E X. Hence ~(T~.~ ) = ~ ( u ( T# ) ) , where a is determined by 0. Therefore a (Tr

= Tr ~, + C, C E fib". This com-

pletes the proof. Corollary. FOral ~ Homeo(%), there exists a inAut(J(PC[i])) satisfying a ( T § T#o~, + C for any 4 in PC[x] if and only if a! is orientation preserving and a a ( 1 ) = 1.

=

From the discussion about PCD] above, we have seen that the maximal ideal of PC[~y~g" can be imagined as "~ - {1 } with a little attached circle at the point 1. It is easy to see that the maximal ideal X, of PC[,]/ffffis "~- {zl, "'", z, } with n little attached circles at Zl, " " , z, respectively. So X, and ~ are also homeomorphie. Henee we can obtain a result similar to Theorem 1 in the suitable statement, and can prove: Theorem 3. For a E Homeo('~), there is a in Aut(J(PC[,])) satisfying a ( T # ) = T§

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SCIENCE IN CHINA (Series A)

Vol. 44

+ C for ~ E PC[.] if and only if a is orientation preserving and a { z l , "", z~ t = { zg(1 ) , "~ zp(n~ } , where p is a permutation of {1, 2 , "", n} with p(k + 1) _----p(k) + l ( m o d n ) , (k = 1,2,

"", n ) .

Acknowledgements The first author would like to thank Profs. Jiaxing Hong, Xiaoman Chen and Kunyu Guo for their encouragement and help. This work was supported by the National Natural Science Foundation of China ( Grant Nos. 19971061 and 19631070) and Funds for Young Fellow of Sichuan University, the Natural Science Foundation of Guangxi.

References 1. 2. 3. 4. 5. 6. 7. 8. 9.

Douglas, R. G . , Banach Algebra Techniques in Operator Theory, New York: Academic Press, 1972. Yah Congquan, Sun Shunhua, Automorphism group on the Toeplitz algebras, Chinese Science Bulletin, 1996, 41(8) : 617. Guo Kunyu, Indices of Toeplitz tuples on pseudoregular domains, Science in China, Ser. A, 2000, 43( 12): 1258. Guo Kunyu, Essentially commutative C" -algebras with essential spectrum homeomorphic to S2"-~ , J. Austral. Math. Soc., Ser. A, 2001, 70: 199. Guo Kunyu,. Indices, characteristic numbers and essential commutants of Toeplitz operators, Ark. Mat., 2000, 38 : 97. Muhly, P. S . , Xia Jingbo, Automorphisms of the Toeplitz algebra, Proc. Amer, Math. Soc., 1992, 116(4): 1067. Douglas, R. G . , Banach algebra techniques in the Toeplitz operators, in Regional Conference in Mathematics, No. 15, Conference Board of the Mathematics Sciences, Rhode Island: A. M. S . , 1972. Bottcher, A . , Silbermann, B . , Analysis of Toeplitz Operators, Berlin: Akademie-Verlag, 1989. Brown, L. G . , Douglas, R. G . , Fillmore, P. A. , Extensions of C" -algebras and K-homology, Annals of Math., 1977, 1051 265.