NEW ZEALAND JOURNAL OF MATHEMATICS
Volume 32 (2003), 183-193
G E N E R ALIZE D P O P O V IC IU F U N C T IO N A L EQUATIO NS IN
B A N A C H M O DU LES O V E R A C *-A L G E B R A A N D
A P P R O X IM A T E A L G E B R A H O M O M O R P H ISM S
C
hun-
G
il
Pa r k
(Received March 2002)
Abstract. W e prove the Hyers-Ulam —Rassias stability of generalized Popoviciu functional equations in Banach modules over a unital C * -algebra.
It is
applied to show the stability of Banach algebra homomorphisms between Ba­
nach algebras associated with generalized Popoviciu functional equations in
Banach algebras.
1. In trod u ction
Let E i and E 2 be Banach spaces with norms ||•||and ||•||, respectively. Consider
/ : E\ —> E 2 to be a mapping such that f { t x ) is continuous in t G M for each fixed
x G Ei. Assume that there exist constants e > 0 and p G [0,1) such that
\\f(x + y) - f ( x ) ~ f(y)\\ < £(|kir + |M|P)
for all x, y G E\. Th.M. Rassias [11] showed that there exists a unique M-linear
mapping T : Ei —> E 2 such that
||/(*)-T(x)|| < 2 ^ l l * r
for all x E Ei.
Throughout this paper, let A be a unital C *-algebra with norm | • |, U{A) the
unitary group of A , A in the set of invertible elements in A, A sa the set of self-adjoint
elements in A, Ai = {a G A | |a| = 1}, and A the set of positive elements in A\.
Let a B and \C be left Banach A-modules with norms || • || and || • ||, respectively.
Let n and k be integers with 2 < k < n — 1.
Recently, T. Trif [19] generalized the Popoviciu functional equation
2000 AM S Mathematics Subject Classification: Primary 47B48, 39B72, 46L05.
Key words and phrases: Banach module over C * -algebra, stability, real rank 0, unitary group,
generalized Popoviciu functional equation, approximate algebra homomorphism.
This work was supported by grant No. 1 9 9 9 -2 -1 0 2 -0 0 1 -3 from the interdisciplinary Research
program year of the K O S E F .
184
CHUN-GIL PARK
Lemma A . [19, Theorem 2.1] Let V and W be vector spaces.
/ : V —> W with /(0 ) = 0 satisfies the functional equation
n-
+ n~2Ck-l ^
n-2Ck-2f
'
A mapping
f (Xj)
i=l
= *• E
1 <Z1
<n
/ (— i ' '
<A>
'
for all x \, •••, x n G V if and only if the mapping / : V —►W satisfies the additive
Cauchy equation / ( x + y) = f ( x ) + f ( y ) for all x, y G V.
The following is useful to prove the stability of the functional equation (A).
Lemma B. [10, Theorem 1] Let a
than 2. Then there are m elements
G
A and |a| < 1 — ^ for some integer m greater
•••, um
G
U(A) such that ma = uH------ bwm.
The main purpose of this paper is to prove the Hyers-Ulam-Rassias stability of
the functional equation (^4) in Banach modules over a unital (7*-algebra, and to
prove the Hyers-Ulam-Rassias stability of algebra homomorphisms between Banach
algebras associated with the functional equation (^4).
2. Stability of Generalized Popoviciu Functional Equations in Banach
Modules over a C*-Algebra Associated with its Unitary Group
We are going to prove the Hyers-Ulam-Rassias stability of the functional equa­
tion ( A ) in Banach modules over a unital (7*-algebra associated with its unitary
group.
For a given mapping / : aB —►aC and a given a G A, we set
D af { x i, ■•• ,x „ ) := n •n- 2C k- 2aj
£
+ n. 2C k- 1 ^ a f ( x i) - k -
f f a *u ± __ + ax±\
i=l
for all
x i , ■■ ■ , x n G
a
B.
Theorem 2.1. Let q =
an^ r = ~ n ^ k ' ^et f : aB
aC be a mapping
with / ( 0) = 0 for which there exists a function
: a B71
[0 , oo) such that
OO
<p(xi,-- - ,x „ )
I , " ’ ,QJx n)
3=0
<
oo\\Duf ( x 1 : --- ,x n)|| <
for all u G U(A) and all x \ , - - - , x n G a&-
,x „ )
(2.i)
Then there exists a unique A-linear
mapping T : aB —►aC such that
\\f{x)-T(x)\\ <
for all x G
a
B.
-- ----- l— ---- tp(qx,rx,--- ,rx )
K•n-lO fc-i
(2.ii)
185
GENERALIZED POPOVICIU FUNCTIONAL EQUATIONS
P r o o f. Put u = 1 G U(A). By [19, Theorem 3.1], there exists a unique additive
mapping T : a& —►aC satisfying (2 .ii). The additive mapping T : a&
aC was
defined by
T(x) =
lim q~3f ( q3x)
j —>oc
for all x G a&By the assumption, for each u G U(A),
q~3\\Duf ( q Jx , - - - ,qJx)\\ < q~3p(q3x , - ■■ ,q3x)
for all x G a&, and
q~3 \\Duf ( q 3x,
,qJx)\\ -*■ 0
as n —> oo for all x G a&- So
D uT(x, ••• , x) = lim q~j D uf ( q 3x, ■■■ , q3x) = 0
j —>oo
for all u G U(A) and all x G a&- Hence
D uT(x, ••• , x) = n •n- 2C k - 2u T ( x ) + n ■n- 2C k- \uT( x) - fc •nCkT(ux) = 0
for all u G U{A) and all x G a&- So
u T( x ) =
T(ux)
for all u G U{A) and all x G a&Now let a G A (a ^ 0) and M an integer greater than 4|a|. Then
a
1
|a|
1
lM l = M |a|<^
= 4 <
2
1
_ 3 = 3
By Lemma B, there exist three elements u i , u 2, u3 G U(A) such that 3-^ = ui +
W2 + W3 . And T (x) = T (3 •
= 3T (|x) for all x G a#- So T (|a:) = |T(x) for
all x E aB- Thus
T(ax)
( ^ - - 3 ^ - x ] = M - t ( \ ■3 - % ) \3
M J
\3
M J
=
t
=
^ - T ( u i x + u2x 4- u3x) =
=
y
=
aT(x)
o
M .
for all x G
a #-
Srri. .
o
M
~ T (3-^x)
3
\ M J
(T(uix) + T ( u 2x ) + T ( u 3x ))
„ a rri/ .
O i + u2 + u3)T( x ) = — •3 — T (x )
Obviously, T(0x) — 0T(x) for all x G a&- Hence
T(ax + by) = T(ax) + T(by) = aT{x) + bT(y)
for all a,b G A and all x, y G a ^>- So the unique additive mapping T :
an A-linear mapping, as desired.
a & —> a C is
□
Applying the unital C*-algebra C to Theorem 2.1, one can obtain the following.
186
CHUN-GIL PARK
C orolla ry 2 . 2 . Let E\ and E 2 be complex Banach spaces with norms || • || and
||•||, respectively. Let q =
and r = —
Let f : E\ —> E 2 be a mapping
with / ( 0) = 0 for which there exists a function (p : E ” —> [0, oo) such that
OO
( p ( x !,• • ■ , X n ) : =
l , - * * , 9 J^ n )
J=0
<
oo II-Da /O t i , - -
- , x „)|| <
for all A G T 1 : = {A G C | |A| = 1} and all x\, ■•■ , x n € E\.
unique C-linear mapping T : E\ —> E 2 such that
, x n)
Then there exists a
l ---- (p(qx,rx, ••• , rx)
||/(x)-T(x)|| < ------- —
for all x £ Ei.
T h eorem 2.3. Let q =
and r = —
f '■a B —>
fee a continuous
mapping with / ( 0) = 0 /o r which there exists a function </? :
—> [0, oo) satisfying
(2.i) such that
\\Duf { x i , - - - ,x n)|| < <^(xi,--- ,x n)
/o r all u G Z7(A) and all xi, ••• ,x n G
7/ £/ie sequence {q~3f ( q 3x ) } converges
uniformly on a &, then there exists a unique continuous A-linear mapping T :
a B —> a C satisfying (2 Ai).
P ro o f. Put u — 1 G U(A). By Theorem 2.1, there exists a unique A-linear map­
ping T : a& —>
satisfying (2.m). By the continuity of / , the uniform convergence
and the definition of T, the A-linear mapping T : a& —> aC is continuous, as de­
sired.
□
T h eorem 2.4. Let q =
and r = —
with / ( 0) = 0 for which there exists a function ip :
/ : a#
a C be a mapping
—■>[0, oo) such that
OO
^ (x i, ••• ,x „ ) : = y ^ g V ( g ~ JXi, •■• ,q~3x n) < ooi
3=0
\\Duf ( x i , ••• ,x n)|| < </?(xi, ••• ,x n)
for all u G U(A) and all x i,-- - ,x n G
mapping T : a& —►
swc/i that
(2.iii)
Then there exists a unique A-linear
||/(x)-T(x)|| < ----- i ---- <p(x,rx, ••• ,rx)
(2.iv)
n —2 ^ fc —1
/o r
all x G
P ro o f. Put « = 1 G U(A). By [19, Theorem 3.2], there exists a unique additive
mapping T : aB —► satisfying (2 iv ). The additive mapping T :
was
defined by
T (x) = lim q3f { q ~ 3x)
j —>oo
for all x E a B.
By the assumption, for each w G £V(A),
9J||7>u/(9_J^ , ' •• ,<7- J x)|| <
qJip(q~Jx, ■■■ , q~3x)
GENERALIZED POPOVICIU FUNCTIONAL EQUATIONS
for all x G
and
qJ\\Duf{q~Jx , - " , q~3x )||
as n
187
0
—►oo for all x € a&- S o
D uT ( x , ••• , x) = lim qj D uf ( q ~ j x, ••• , q~j x) = 0
j —>OG
for
all u G U (A ) and all x € aB. Hence
D uT(x,
, x) = n •n- 2C k - 2uT(x) + n •n_ 2Cfc_inT(x) - fc •nC kT(ux) = 0
for all u G W(-A) and all x G ,4# . So
uT(x) = T (u x )
for all w G
and all x G a # The rest of the proof is the same as the proof of Theorem 2.1.
□
3. Stability of Generalized Popoviciu Functional Equations in Banach
Modules over a C * —Algebra
Given a locally compact abelian group G and a multiplier co on G, one can
associate to them the twisted group C*-algebra C * ( G , lj). (7*(Zm,u;) is said to be
a noncommutative torus of rank m and denoted by A w. The multiplier w determines
a subgroup Sw of G, called its symmetry group, and the multiplier is called totally
skew if the symmetry group Sw is trivial. And A w is called completely irrational if
u) is totally skew (see [2 ]). It was shown in [2 ] that if G is a locally compact abelian
group and a; is a totally skew multiplier on G, then C*( G, u>) is a simple C*-algebra.
It was shown in [3, Theorem 1.5] that if A w is a simple noncommutative torus then
A w has real rank 0 , where “real rank 0” means that the set of invertible self-adjoint
elements is dense in the set of self-adjoint elements (see [3], [6 ]).
From now on, assume that ip : a Bn —►[0, oo) is a function satisfying (2.i), and
that q =
and r = - ^ .
We are going to prove the Hyers-Ulam-Rassias stability of the functional equa­
tion (A) in Banach modules over a unital C'*-algebra.
Theorem 3.1. Let q be not an integer.
/ ( 0) = 0 such that
Let f : a& —►aC be a mapping with
\ \ D i f ( x ! , - - - , £ n )|| <
<fi(Xi, ■ ■ ■ , X n )
for all x i, ••• , x n G a&- Then there exists a unique additive mapping T : a& —> aC
satisfying (2 .ii). Further, if f(Xx) is continuous in A g I for each fixed x G a&,
then the additive mapping T : a& —» aC is K.-linear.
Proof. By the same reasoning as the proof of Theorem 2.1, there exists a unique
additive mapping T : a& —> aC satisfying (2 .ii).
Assume that /(A x ) is continuous in A G M for each fixed x G a&- By the
assumption, q is a rational number which is not an integer. The additive mapping
T given above is similar to the additive mapping T given in the proof of [11,
Theorem]. By the same reasoning as the proof of [11, Theorem], the additive
mapping T : a& —> aC is M-linear.
□
188
CHUN-GIL PARK
T h eorem 3.2. Let q be not an integer. Let f :
with / ( 0) = 0 such that
a&
—> a C be a continuous mapping
\\Daf { x i , - - ’ , x n)\\ <
, x n)
for all a G A f U { i } and all
, x n G a &- If the sequence { q ~ i f ( q 3x ) } con­
verges uniformly on a &, then there exists a unique continuous A-linear mapping
T : a & —*■a C satisfying (2 .ii).
P ro o f. Put a = 1 G A f . By Theorem 3.1, there exists a unique M-linear mapping
T : a& —+ aC satisfying ( 2.ii ). By the continuity of / and the uniform convergence,
the IR-linear mapping T : a&
aC is continuous.
By the same reasoning as the proof of Theorem 2.1,
T(ax) = a T( x )
for all a G A± U {*}.
For any element a
G
A, a = g-^ ° - + i q~ f , and
elements, furthermore, a — ( a\ a ) + — ( a+2a )
( a~^Q ) + , ( a+2 ) , ( a^
38.8]). So
) + , and ( a~ “ )
/ 'a + a * \ +
f a + a*\
T(ax) = T [ ( — - — I x — ( — - — I
T(x)+
a + a*\ + r^, .
+
<l-^ f ■
are self-adjoint
f ) + — i { a~2i )
> where
are positive elements (see [4, Lemma
. f a — a*\ +
. ( a — a*
x + i [ ———
x —i
2i J
V 2i
T (-x)+
f a + a*\
and
rri/ .
— —
T(ix)
. f a — a*x +
+
)
T(x)
T(x)
i
2
+
i {— ) - ' [ T * - ) ) T(X)
— aT(x)
for all a
G
A and all x G
a
&-
Hence
T(ax + by) = T(ax) + T(by) = aT(x) + bT(y)
for all a,b
G
A and all x, y G a&, as desired.
□
T h eorem 3.3. Let A be a unital C* -algebra of real rank 0, and q not an integer.
Let f : a & —» a C be a continuous mapping with f ( 0) = 0 such that
\\Daf ( x i , - - - , x n)\\ < (p(xi, ■■■ , x n)
GENERALIZED POPOVICIU FUNCTIONAL EQUATIONS
189
for all a G (A+ fl Ain) U { i } and all x\, ••• ,x „ G a B. If the sequence {q~3 f { q3x )}
converges uniformly on a B, then there exists a unique continuous A-linear mapping
T : a & ~> a C satisfying (2 .ii).
P r o o f. By the same reasoning as the proof of Theorem 3.2, there exists a unique
continuous M-linear mapping T : a B —* a C satisfying (2 .ii), and
T(ax) = aT(x)
(3.1)
all a G ( A f fl Ain) U {z} and all x G aB.
Let b G A~l \ A in. Since Ain fl A sa is dense in A sa, there exists a sequence { 6m}
in Ain n A sa such that bm —> b as m —> oo. Put cm = j ^ b m. Then cm —> ^ b = b
for
as m —■> oo. Put am = y/c^cm . Then am —> b as m —> oo and am G A^ fl Ain.
Thus there exists a sequence {a m} in A~{ fl Ain such that am —> b as m —> oo, and
by the continuity of T
lim T(amx) = T{ lim amx) = T(bx)
m —►oo
for
(3.2)
m —►oo
all x G aB. B y (3.1),
||T(amx) - bT(x) ||= ||amT(x) - bT{x) || -
as m
||MT(x) - bT{x) ||= 0
(3.3)
—►oo. By (3.2) and (3.3),
||T(te) - bT(x )II
for all x G aB.
<
||r(te) - T(amx )II + IIT(amx) - bT(x)\\
—>
0
as
m —* oo
(3.4)
By (3.1) and (3.4), T(ax) = aT(x) for all a G A~l U { i } and all
x G a B.
The rest of the proof is similar to the proof of Theorem 3.2.
T h eorem 3.4. Let q be not an integer.
/ ( 0) = 0 such that
\ \ D a f { x i , ’ - ’ ,®n)||
Let f : a B
<
aC
□
be a mapping with
i p { X ! , - ■■ , X n )
for all a G A f U { i } and all x \,••• , x n G a B. If f{ Xx) is continuous in X G M
for each fixed x G a B, then there exists a unique A-linear mapping T : a B —> a C
satisfying (2 .ii).
P r o o f. Put a = 1 G A f . By Theorem 3.1, there exists a unique R-linear mapping
T : aB —» aC satisfying (2 .ii).
The rest of the proof is similar to the proof of Theorem 3.2.
□
T h eorem 3.5. Let A be a unital C* -algebra of real rank 0, and q not an integer.
Let f : a & —> a C be a mapping with /( 0 ) = 0 such that
\\Daf ( x i , - " , x n)\\ < <p(xi , - - - , x n)
for all a G (A+ n A*n) U { i } and all X\,--- , x n G a B. Assume that f{ax) is
continuous m a G i i U l for each fixed x G a B, and that the sequence {q~i f ( q 3x )}
converges uniformly on A\ for each fixed x G a B. Then there exists a unique
A-linear mapping T : a B —> a C satisfying (2 .ii).
190
CHUN-GIL PARK
Proof. By the same reasoning as the proof of Theorem 3.2, there exists a unique
M-linear mapping T : aB —* aC- satisfying (2 .ii), and
T (a x ) = aT(x)
for all a G ( A f fl Ain) U {«} and all x G a B . By the continuity of / and the uniform
convergence, one can show that T(ax) is continuous in a G A\ for each x G aB.
The rest of the proof is similar to the proof of Theorem 3.3.
□
Theorem 3.6. Let f : aB
aC be a mapping with /(0 ) = 0 such that
\\Daf ( x i , ••• ,x„)|| < <p(xi, ••• , x„ )
for all a G A^ U { i } U l and all
- , x n G aB.
A-linear mapping T : aB —> aC satisfying (2 .ii).
Then there exists a unique
Proof. By the same reasoning as the proof of Theorem 2.1, there exists a unique
additive mapping T : aB —►aC satisfying (2 .ii), and
T(ax) = aT(x)
for all a G A f U {i}U lR and all x G a $- So the mapping T : a $ —^aC is M—linear,
and satisfies
T(ax) = aT(x)
for all a G A * U { i } and all x G aB-
The rest of the proof is the same as the proof of Theorem 3.2.
Similarly, for the case that <p : a B 71
□
[0, oo) is a function such that
OO
Y , q 3(p(q~3xi , ' - - , q~3x n) < oo
3=0
for all xi, ■■• , x n G aB, one can obtain similar results to the theorems given above.
4. Stability of Generalized Popoviciu Functional Equations in Banach
Algebras and Approximate Algebra Homomorphisms Associated
with (A)
In this section, let A and B be Banach algebras with norms || • || and || • ||,
respectively.
D.G. Bourgin [5] proved the stability of ring homomorphisms between Banach
algebras. In [1], R. Badora generalized the Bourgin’s result.
We prove the Hyers-Ulam -Rassias stability of algebra homomorphisms between
Banach algebras associated with the functional equation (A).
Theorem 4.1. Let A and B be real Banach algebras, and q not an integer. Let
f : A —>B be a mapping with / ( 0) = 0 for which there exists a function ip : A x A —►
[0, oo) such that
OO
$ ( x , y ) ;= Y L q~3^ qJx^y ) <
oo,
(4.i)
3=0
\\D\f(x\, ■•• , ar„)|| < (p{x\, ••* , x n),
\\f(x-y) - f(x)f(y)\\ < ^ { x , y )
(4-ii)
(4.iii)
GENERALIZED POPOVICIU FUNCTIONAL EQUATIONS
191
for all x, y, x\, ■•• , x n G A . If / ( Ax) is continuous in A G M for each fixed x G A ,
then there exists a unique algebra homomorphism T : A —> B satisfying (2 .ii).
Further, if A and B are unital, then f itself is an algebra homomorphism.
P ro o f. Under the assumption (2 .i) and (4 .ii), in Theorem 3.1, we showed that
there exists a unique E-linear mapping T : A —> B satisfying (2 .ii). The IR-linear
mapping T : A —* B was given by
T(x) =
for all x
G
lim q~3f ( q 3x)
j —►oo
A. Let
R(x,y) =
for all x , y G A.
f ( x -y) - f { x ) f ( y )
By (4 .i), we get
lim q~~3R(q3x , y )
= 0
j —1-oo
for all x , y
G
A. So
T( x •y)
f ( q J{ x - y ) )
—
=
=
for all x , y
G
lim q~ J
(4-1)
j —*oo
lim q~Jf ( { q Jx)y)
j —>oo
lim q~3 ( f{qJx ) f { y ) + R(qJx , y ) ) = T ( x ) f { y )
J—>oo
(4.2)
A. Thus
T { x ) f ( q Jy) = T( x{ q3y)) = T((q3x)y) = T(q3x ) f ( y ) = q3T ( x ) f ( y )
for all x, y
G
A. Hence
T(x)q~3f ( q 3y) = T { x ) f ( y )
for all x, y
G
(4.3)
A. Taking the limit in (4.2) as j —> oo, we obtain
T( x ) T( y) = T ( x ) f ( y )
for all x , y
G
A. Therefore,
T ( x ■y ) = T( x) T( y )
for all x, y G A. So T : A —>B is an algebra homomorphism.
Now assume that A and B are unital. By (4.2),
T(y) =
for all y
G
T( 1 •y) = T ( l ) f ( y ) = f ( y )
A . So / : A —>B is an algebra homomorphism, as desired.
□
T h eorem 4.2. Let A and B be complex Banach algebras. Let f : A —> B be a
mapping with / ( 0) = 0 for which there exists a function ^ : A x A —> [0, oo)
satisfying (4 .i) and (4 .iii) such that
\\D\f(xi,--- , x n)\\ < < p ( x i , - - - , x n)
for all A G T 1 and all X\, ••• , x n G A.
(4.iv)
Then there exists a unique algebra homo­
morphism T : A —> B satisfying ( 2.ii). Further, if A and B are unital, then f itself
is an algebra homomorphism.
192
CHUN-GIL PARK
Proof. Under the assumption (2 .i) and (4 .iv), in Corollary 2.2, we showed that
there exists a unique C-linear mapping T : A —>B satisfying (2 .ii).
The rest of the proof is the same as the proof of Theorem 4.2.
□
Theorem 4.3. Let A and B be complex Banach * - algebras. Let f : A —> B be
a mapping with / ( 0 ) = 0 for which there exists a function
: A x A —> [0, oo)
satisfying (4 .i) and (4 .in) such that
\\D\f(xi, ■■■ ,x n)|| <
, x n)
\\f(x*)~f{x)*\\ < < p( x , - - - , x )
(4.iv)
(4.v)
for all A G T 1 and all x , x \ , - - - , x n G A. Then there exists a unique *-algebra
homomorphism T : A —> B satisfying ( 2 .ii). Further, if A and B are unital, then
f itself is a * -algebra homomorphism.
Proof. By the same reasoning as the proof of Theorem 4.2, there exists a unique
C-linear mapping T : A —►B satisfying (2 .ii).
Now
q~3 \\f{q3x*) - f( q3x)*\\ < q~3<p{qJx, ••• , q3x)
for all x
G
A. Thus
0
q~3 \\f{q3x*) - f { q 3x)*\\
as n —» oo for all x
G
A. Hence
T(x*) = lim q~3f { q 3x*) = lim q~3f ( q 3x)* = T ( x ) *
j —>oo
j —>oc
for all x G A.
The rest of the proof is the same as the proof of Theorem 4.2.
□
Similarly, for the case that ip : A n —> [0, oo) and if) : A x A —►[0, oo) are functions
such that
OO
^ 2 q 3v { q ~ 3x i , ■- , q~3x n)
<
oo,
<
oo
3=0
OO
^ q 3^ { q ~ 3x , y )
3=0
for all x , y , x i ,- - - , x n
above.
G
A, one can obtain similar results to the theorems given
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Chun-Gil Park
Department of Mathematics
Chungnam National University
D aejeon 3 05 -7 6 4
KOREA
cgpar kmat h.cnu.ac.kr