SCHWARZ AND GRÜSS TYPE INEQUALITIES FOR C*-SEMINORMS AND POSITIVE LINEAR FUNCTIONALS ON BANACH -MODULES
AMIRG.GHAZANFARIANDSEVERS.DRAGOMIR
Abstract. Let be a unital Banach -algebra, a C -seminorm or a positive A linear functional on and X be a semi-inner product -module. We de…ne A A a real function on X by (x) = ( ( x; x ))1=2 and show that the Schwarz h i inequality holds, therefore (X; ) is a semi-Hilbert -module. We also obtain A some Grüss type inequalities for C-seminorms and positive linear functionals on . A
1. Introduction In 1934, G. Grüss [6] showed that for two Lebesgue integrable functions f; g : [a; b] R, ! 1 b 1 b 1 b 1 f(t)g(t)dt f(t)dt g(t)dt (M m)(N n); b a a b a a b a a 4 Z Z Z provided m; M; n; N are real numbers with the property < m f M < 11 1 and < n f N < a.e. on [a; b]: The constant 4 is best possible in the sense 1 that it cannot be replaced1 by a smaller constant. The following inequality of Grüss type in real or complex inner product spaces is known [3].
Theorem 1. Let (H; ; ) be an inner product space over K (K = C; R) and h i e H; e = 1. If ; ; ; K and x; y H are such that conditions 2 k k 2 2 Re e x; x e 0; Re e y; y e 0 h i h i or, equivalently, + 1 + 1 x e ; y e 2 2j j 2 2j j
hold, then we have the inequality 1 (1.1) x; y x; e e; y : jh i h i h ij 4j jj j 1 The constant 4 is best possible in (1.1). Dragomir in [5] presented re…nements of the Grüss type inequality (1.1) and some companions and applications. Iliševi´c and Varošane´c in [7] have proved a re…nement of a Grüss type inequality in proper H-modules and C-modules.
2000 Mathematics Subject Classi…cation. Primary 46L08, 46H25; Secondary 26D99, 46C99. Key words and phrases. Inner product C -modules, inner product -modules, Schwarz inequal- ity, Grüss inequality, C-seminorms. 1 2 GHAZANFARIANDDRAGOMIR
In this paper we obtain a version for the Schwarz inequality and provide inequal- ities of Grüss type in inner product Banach -modules.
2. Preliminaries Let be a -algebra. A seminorm on is a real-valued function on such that A A A for a; b and C: (a) 0; (a) = (a); (a + b) (a) + (b).A 2 A 2 j j seminorm on is called a C-seminorm if it satis…es the C-condition: (aa) = 2 A ( (a)) (a ): By Sebestyen’stheorem [2, Theorem 38.1] every C-seminorm on a -algebra2 A is submultiplicative, i.e., (ab) (a) (b)(a; b ), and by [1, A 2 A Section 39, Lemma 2 (i)] (a) = (a). For every a , the spectral radius of a is de…ned to be r(a) = sup : (a) . 2 A The Pták function fjonj -algebra2 A gis de…ned to be : [0; ), where 1=2 A A! 1 (a) = (r(aa)) . This function has important roles in Banach -algebras, for example, on C-algebras, is equal to the norm and on hermitian Banach -algebras is the greatest C-seminorm. By utilizing properties of the spectral radius and the Pták function, V. Pták [9] showed in 1970 that an elegant theory for Banach -algebras arises from the inequality r(a) (a). This inequality characterizes hermitian (and symmetric) Banach -algebras, and further characterizations of C-algebras follow as a result of Pták theory. Let be a -algebra. We de…ne + by A A n + = akak : n N; ak for k = 1; 2; :::; n ; A 2 2 A (k=1 ) X and call the elements of positive. The set + of positiveA elements is obviously a convex cone (i.e., it is closed under convex combinationsA and multiplication by positive constants). Hence we call + the positive cone. By de…nition, zero belongs to +. It is also clear that eachA positive element is hermitian. A
De…nition 1. Let be a -algebra. A semi-inner product -module (or semi- inner product -module)A is a complex vector space which is alsoA a right -module X with a sesquilinear semi-inner product ; : X X , ful…lling A h i !A x; ya = x; y a for x; y X; a ; (right linearity) h i h i 2 2 A x; x + for x X: (positivity) h i 2 A 2 Furthermore, if X satis…es the strict positivity condition x = 0 if x; x = 0; (strict positivity) h i then X is called an inner product -module (or inner product -module). A Let be a seminorm or a positive linear functional on and (x) = ( ( x; x ))1=2 (x X). If is a seminorm on a semi-inner product A-module X, thenh(X; i ) is said2 to be a semi-Hilbert -module. A If is a norm on anA inner product -module X, then (X; ) is said to be a pre-Hilbert -module. A A pre-HilbertA -module which is complete with respect to its norm is called a Hilbert -module.A A SCHWARZANDGRÜSSTYPEINEQUALITY 3
3. Schwarz Inequality Let ' be a positive linear functional on a -algebra and X be a semi-inner -module. We can de…ne a sesquilinear form on X X byA (x; y) = ' ( x; y ); the ASchwarz inequality for implies that h i
(3.1) ' ( x; y ) 2 ' ( x; x ) ' ( y; y ) : j h i j h i h i Therefore (x) = ( ( x; x ))1=2 is a seminorm on X and (X; ) is a semi-Hilbert -module. h i A Proposition 1. Let be a -algebra and X be a semi-inner product -module. If A 1=2 A is a C-seminorm on and (x) = ( ( x; x )) (x X) then the Schwarz inequality holds, that is A h i 2
(3.2) ( ( x; y ))2 ( x; x ) ( y; y ) : h i h i h i Therefore (x + y) (x) + (y) for every x; y X. Thus is a seminorm on X and (X; ) is a semi-Hilbert -module. 2 A Proof. First we show that has the following monotone property: (a) (b) if 0 a b (a; b ): 2 A Let J be the ideal de…ned by
J = a : (a) = 0 f 2 A g and ~ the algebra norm de…ned on the algebra =J by A ~(b) = (a)(a b =J ): 2 2 A We denote by B the completion of =J with respect to the norm ~ and we A denote also by ~ the usual extension of this norm to B . Suppose that a; b 2 A and 0 a b, then [a] = a + J b + J = [b] in =J . Since ~ is a C-norm on A B therefore ~([a]) ~([b]) [8, Theorem 2.2.5 (3)] consequently (a) (b). The rest of the proof is similar to the proof of Lemma 15.1.3 in [10]; we include it for the sake of completion: If ( x; y ) = 0, this is trivial. Suppose that ( x; y ) = 0 put a := x; x , b := y;h y , ci:= x; y and let be an arbitrary real number.h i 6 Then h i h i h i 2 (3.3) 0 x yc; x yc = a 2cc + cbc: h i Adding the self-adjoint element 2cc at both sides and taking seminorms, we obtain
2 2 2 2 2 2 (c) (a + cbc) (a) + (cbc) (a) + (c) (b): j j Now, for all R we have the inequality 2 0 (c)2 (b) 2 2 (c)2 + (a); j j so that the discriminant 4 (c)4 4 (c)2 (b) (a) must be non-positive. Since we suppose that (c) = 0, by dividing with 4 (c)2 we deduce the desired inequality (3.2). 6 4 GHAZANFARIANDDRAGOMIR
It follows that (x + y)2 = ( x + y; x + y ) h i ( x; x ) + ( x; y ) + ( y; x ) + ( y; y ) h i h i h i h i 1 1 ( x; x ) + 2( ( x; x )) 2 ( ( y; y )) 2 + ( y; y ) h i h i h i h i 1 1 2 = ( ( x; x )) 2 + ( ( y; y )) 2 h i h i = ( (x) + (y))2:
Therefore is a seminorm on X, and (X; ) is a semi-Hilbert -module. A Example 1.
(a) Let be a -algebra and a positive linear functional or a C-seminorm on A. It is known that ( ; ) is a semi-Hilbert -module over itself with A A A the inner product de…ned by a; b := ab, in this case = . (b) Let be a hermitian Banachh -algebrai and be the Pták function on . If X isA a semi-inner product -module and P (x) = (( x; x ))1=2 (x AX), then (X;P ) is a semi-HilbertA -module. h i 2 A (c) Let be a A-algebra and be the auxiliary norm on . If X is an innerA product -module andj x j = x; x 1=2 (x X), thenA (X; ) is a pre-Hilbert -module.A j j j h i j 2 j j A Remark 1. Let ' be a positive linear functional on a unital Banach -algebra , X a semi-inner -module and x; y X. Put a := x; x ; b := y; y and c := x; yA. From (3.3) and A[1, Section 37 Lemma2 6 (iii)] we haveh i h i h i
2 2 2 2cc a + cbc; therefore 2'(cc) '(a + cbc) = '(a) + '(cbc) 2 '(a) + '(cc)r(b): 2 Thus for all R inequality 0 '(cc)r(b) 2'(cc) + '(a) holds. So the dis- 2 2 criminant '(cc) '(cc)'(a)r(b) = '(cc)('(cc) '(a)r(b)) 0. This implies that
(3.4) '(cc) '(a)r(b) or '( x; y y; x ) '( x; x )r( y; y ): h i h i h i h i Now suppose that X is a C-module on C-algebra and a; b; c are as above. By A 2 [8, Theorem 3.3.6] there is a state ' on such that '(cc) = cc = c . Using inequality (3.4) we have A k k k k
2 c = '(cc) '(a)r(b) a b : k k k kk k Therefore (3.4) is a re…nement of Schwarz’s inequality for C-modules [10, Lemma 15.1.3].
4. Grüss Type Inequalities We assume, unless stated otherwise, throughout this section that is a unital Banach -algebra. The following Lemma 1 is a version of [4, Lemma 2.1]A for a semi- inner product -module and the following Lemma 3 is a version of ([4, Lemma 2.4]) for an inner productA -module. A SCHWARZANDGRÜSSTYPEINEQUALITY 5
Lemma 1. Let X be an semi-inner product -module, and x; y X; ; C. Then A 2 2 Re y x; x y 0 h i if and only if + + 1 x y; x y 2 y; y : 2 2 4j j h i Proof. Follows from the equalities: 1 Re y x; x y = ( y x; x y + x y; e x ) h i 2 h i h i + + + = y; x y; y x; x + x; y 2 h i 2 h i h i 2 h i 1 + + = 2 y; y x y; x y : 4j j h i 2 2 Lemma 2. Let X be an inner product -module and x; y; e X. If e; e is idempotent, then e e; e = e, and thereforeA 2 h i h i e; e e; x = e; x ; x; e = x; e e; e : h i h i h i h i h i h i Proof. Observe that the equality e e; e e; e e; e e = e e; e ; e e; e e e; e ; e e; e e; e + e; e h h i h i i h h i h ii h h i i h h ii h i = e; e e; e e; e e; e e; e e; e e; e + e; e h i h i h i h i h i h i h i h i = 0; implies that e e; e e = 0. h i The rest follows from this fact and we omit the details.
Lemma 3. Let X be an inner product -module and be a C-seminorm or a positive linear functional on and (x)A = ( ( x; x ))1=2 (x X). If x; e X and e; e is an idempotent thenA h i 2 2 h i 0 x; x x; e e; x h i h i h i and ( x; x x; e e; x ) inf (x e)2: h i h i h i C 2 Proof. Observe, for any a , that 2 A x ea; x e e; x = x; x x; e e; x ea; x + ea; e e; x h h ii h i h h ii h i h h ii = x; x x; e e; x a e; x + a e; e e; x h i h i h i h i h i h i = x; x x; e e; x : h i h i h i This implies that x; x x; e e; x = x e e; x ; x e e; x 0: h i h i h i h h i h ii Also observe, for any C, that 2 x e; x e e; x = x; x x; e e; x e; x + e; e e; x h h ii h i h h ii h i h h ii = x; x x; e e; x e; x + e; e e; x h i h i h i h i h i h i = x; x x; e e; x : h i h i h i 6 GHAZANFARIANDDRAGOMIR
Using Schwarz’sinequality, we have ( x; x x; e e; x )2 = ( x e; x e e; x )2 h i h i h i h h ii ( x e; x e ) ( x e e; x ; x e e; x ) h i h h i h ii = ( x e; x e ) ( x; x x; e e; x ); h i h i h i h i therefore giving the bound 2 ( x; x x; e e; x ) ( x e; x e ) = (x e) C: h i h i h i h i 2 Taking the in…mum in the above relation over C, we deduce 2 ( x; x x; e e; x ) inf (x e)2: h i h i h i C 2
Let X be a semi-inner product -module, x; y X; ; C and be a C- seminorm or a positive linear functionalA on . Put2 (x) = ( 2( x; x ))1=2(x X). By Lemma 1, Re y x; x y 0 impliesA that h i 2 h i + 2 + + x y = x y; x y 2 2 2 1 2 ( y; y ) 4j j h i 1 = 2 (y)2: 4j j
Also, let 0 = e X and e; e be idempotent. If is a C-seminorm, then it is trivial that 6 ( e;2 e ) = 0 orh ( ie; e ) = 1, i.e., (e) 1. h i h i Lemma 4. Let X be an inner product -module, be a C-seminorm on and (x) = ( ( x; x ))1=2 (x X). If x; y;A e X, e; e is idempotent and ;A ; ; are real orh complexi numbers2 such that 2 h i + 1 + 1 x e ; y e 2 2j j 2 2j j hold, then one has the inequality 1 ( x; y x; e e; y ) : h i h i h i 4j jj j Furthermore, if there is a non zero element f in X such that e; f = 0 and (f) = 1 h i 6 0; then the constant 4 is best possible. Proof. By (3.2), is a seminorm on X. It can be easily shown that, x; y x; e e; y = x e e; x ; y e e; y : h i h i h i h h i h ii From the Schwarz inequality, we obtain ( x; y x; e e; y ) h i h i h i = ( x e e; x ; y e e; y ) h h i h ii 1 1 (4.1) ( x e e; x ; x e e; x ) 2 ( y e e; y ; y e e; y ) 2 h h i h ii h h i h ii 1 1 = ( x; x x; e e; x ) 2 ( y; y y; e e; y ) 2 : h i h i h i h i h i h i SCHWARZANDGRÜSSTYPEINEQUALITY 7
Using Lemma 3 and the above assumptions, we have that
1 ( x; x x; e e; x ) 2 inf (x e) h i h i h i C 2 + 1 x e 2 2j j and 1 ( y; y y; e e; y ) 2 inf (y e) h i h i h i C 2 + 1 y e : 2 2j j Therefore the desired inequality is obtained. 1 Now we show that the constant 4 is best possible. If f is a non zero element of X with (f) = 0 such that e; f = 0 and given > 0; then for 6 h i + + x = j j f + e; y = j j f + e 2( (f) + ) 2 2( (f) + ) 2 the assumptions of the previous lemma hold and in this case (f)2 ( x; y x; e e; y ) = j jj j : h i h i h i 4 ( (f) + )2 1 Now if c is a constant such that 0 < c < 4 then there is a > 0 such that (f)2 4( (f)+)2 > c. Therefore
( x; y x; e e; y ) > c : h i h i h i j jj j In the following lemma ' is a positive linear functional on . Putting (x) = 1 A '( x; x ) 2 (x X), by (3.1), is a seminorm on X. Therefore we have: h i 2 Lemma 5. Let X be an inner product -module and ' a positive linear functional on . If x; y; e X, e; e is idempotentA and ; ; ; are real or complex numbers suchA that 2 h i + 1 + 1 x e (e); y e (e) 2 2j j 2 2j j hold, then one has the inequality 1 '( x; y x; e e; y ) (e)2: h i h i h i 4j jj j Furthermore, if there is a non zero element f in X such that e; f = 0 and (f) = 0 1 h i 6 then the constant 4 is best possible. We are able now to state our …rst main result:
Theorem 2. Let X be an inner product -module, a C-seminorm on and 1 A A (x) = ( ( x; x )) 2 (x X). If x; y; e X, e; e is idempotent and ; ; ; are real or complexh i numbers2 such that 2 h i + 1 + 1 x e ; y e 2 2j j 2 2j j 8 GHAZANFARIANDDRAGOMIR hold, then one has the inequality
(4.2) ( x; y x; e e; y ) h i h i h i 1 1 1 + 2 2 2 x e 4j jj j 4j j 2 ! 1 1 + 2 2 2 y e : 4j j 2 ! Furthermore, if there is a non zero element f in X such that e; f = 0 and (f) = 0 1 h i 6 then the constant 4 is best possible. Proof. A simple calculation shows that e e e; x ; e e; x e e x; x e = x; x x; e e; x ; h h i h i i h i h i h i h i therefore Re e e e; x ; e e; x e Re e x; x e = x; x x; e e; x : h h i h i i h i h i h i h i Since a; b + b; a 1 a + b; a + b , so h i h i 2 h i 1 Re e e e; x ; e e; x e 2 e; e : h h i h i i 4j j h i As in the proof of Lemma 1 1 + + Re e x; x e = 2 e; e x e; x e ; h i 4j j h i 2 2 therefore + + (4.3) x; x x; e e; x x e; x e : h i h i h i 2 2 Analogously + + (4.4) y; y y; e e; y y e; y e : h i h i h i 2 2 We obtain + 2 + 2 ( x; x x; e e; x ) ( y; y y; e e; y ) x e y e : h i h i h i h i h i h i 2 2 Finally, using the elementary inequality for real numbers (m2 n2)(p2 q2) (mp nq)2 on
1 1 1 + 2 m = ; n = 2 (x e)2 ; 2j j 4j j 2 1 1 1 + 2 p = ; q = 2 (y e)2 ; 2j j 4j j 2 SCHWARZANDGRÜSSTYPEINEQUALITY 9 we get + + x e y e 2 2 1 1 1 + 2 2 2 x e 4j jj j 4j j 2 ! 1 1 + 2 2 2 y e : 4j j 2 ! 1 The fact that 4 is the best constant can be proven in a similar manner to the one in the previous lemma. The details are omitted. Similarly for a positive linear functional ', the following theorem holds. Theorem 3. Let X be an inner product -module, ' a positive linear functional 1 A and (x) = ('( x; x )) 2 (x X). If x; y; e X, e; e is idempotent and ;A ; ; are real orh complexi numbers2 such that 2 h i + 1 + 1 x e (e); y e (e) 2 2j j 2 2j j hold, then one has the inequality
(4.5) '( x; y x; e e; y ) h i h i h i 1 1 1 + 2 2 (e)2 2(e)2 x e 4j jj j 4j j 2 ! 1 1 + 2 2 2(e)2 y e : 4j j 2 ! Furthermore, if there is a non zero element f in X such that e; f = 0 and (f) = 1 h i 6 0; then the constant 4 is best possible. Remark 2. (i) If in the above theorem ' is a state on then, obviously, inequality (4.5) becomes the following: A
(4.6) '( x; y x; e e; y ) h i h i h i 1 1 1 + 2 2 2 x e 4j jj j 4j j 2 ! 1 1 + 2 2 2 y e : 4j j 2 !
(ii) Let X be a C-module and x; y; e X. If e; e is idempotent and ; ; ; are real or complex numbers such2 that h i + 1 + 1 (4.7) x e ; y e 2 2j j 2 2j j
10 GHAZANFARIANDDRAGOMIR
hold, then by [8, Theorem 3.3.6] there are states ' and on a C-algebra such that A '( x; x x; e e; x ) = x; x x; e e; x h i h i h i k h i h i h i k and
( y; y y; e e; y ) = y; y y; e e; y : h i h i h i k h i h i h i k Inequalities (4.3), (4.4) imply that
+ + (4.8) '( x; x x; e e; x ) ' x e; x e h i h i h i 2 2 and
+ + (4.9) ( y; y y; e e; y ) y e; y e : h i h i h i 2 2 From Schwarz’s inequality (4.1) and inequalities (4.8), (4.9) we get
1 1 x; y x; e e; y x; x x; e e; x 2 y; y y; e e; y 2 k h i h i h i k k h i h i h i k k h i h i h i k 1 1 = '( x; x x; e e; x ) 2 ( y; y y; e e; y ) 2 h i h i h i h i h i h i + + x e y e : 2 2 By (4.7) we have
+ + x e x e j j; 2 2 2 + + y e y e j j: 2 2 2
Using the elementary inequality for real numbers
(m2 n2)(p2 q2) (mp nq)2 for
1 1 1 + 2 2 m = ; n = 2 x e 2j j 4j j 2 " # and
1 1 1 + 2 2 p = ; q = 2 y e ; 2j j 4j j 2 " # SCHWARZANDGRÜSSTYPEINEQUALITY 11
we obtain x; y x; e e; y k h i h i h i k 1 1 1 + 2 2 2 x e 4j jj j 4j j 2 ! 1 1 + 2 2 2 y e 4j j 2 ! 1 1 1 + 2 2 2 x e 4j jj j 4j j 2 ! 1 2 2 1 + (4.10) 2 y e ; 4j j 2 !
which is a re…nement of the Grüss inequality for C-modules [7, Theorem 5.1]. (iii) By inequality (3.4) we may obtain another re…nement of [7, Theorem 5.1]: 1 Put G = x; y x; e e; y and R(x) = (r x; x ) 2 . For every positive linear functionalh i ' hon i h wei have h i A 1 2 2 1 1 2 + '(GG) x e 4j jj j 4j j 2 ! 1 1 + 2 2 2 R y e ; 4j j 2 !
and we know that there is a state ' on the C-algebra such that '(GG) = 2 A GG = G . k k k k 5. A Companion of the Grüss Inequality The following companion of the Grüss inequality for positive linear functionals holds: Theorem 4. Let X be an inner product -module, ' a positive linear functional on and x; y; e X. If e; e is idempotentA and ; ; ; are real or complex numbersA such that2 h i Re e x; x e 0; Re e y; y e 0 h i h i hold, then one has the inequality 1 '( x; y x; e e; y ) (e)2 h i h i h i 4j jj j + + e e e; x e e e; x : 2 h i 2 h i Furthermore, if there is a non zero element f in X such that e; f = 0 and (f) = 0 1 h i 6 then the constant 4 is best possible. 12 GHAZANFARIANDDRAGOMIR
Proof. For every k K we have 2 x; x x; e e; x = x ke; x ke ke e e; x ; ke e e; x : h i h i h i h i h h i h ii + For k = 2 Lemma 1 implies that 1 + + x; x x; e e; x 2 e; e e e e; x ; e e e; x : h i h i h i 4j j h i 2 h i 2 h i Therefore 1 + 2 '( x; x x; e e; x ) 2(e)2 e e e; x : h i h i h i 4j j 2 h i Analogously 1 + 2 '( y; y y; e e; y ) 2(e)2 e e e; x : h i h i h i 4j j 2 h i Using Schwarz’sinequality and the elementary inequality for real numbers (m2 n2)(p2 q2) (mp nq)2 we obtain ('( x; y x; e e; y ))2 h i h i h i '( x; x x; e e; x )'( y; y y; e e; y ) h i h i h i h i h i h i 1 + + 2 (e)2 e e e; x e e e; x : 4j jj j 2 h i 2 h i Therefore we get '( x; y x; e e; y ) h i h i h i 1 + + (e)2 e e e; x e e e; x : 4j jj j 2 h i 2 h i Other inequalities related to the Grüss inequality such as Theorem 18, Propo- sition 18, Theorem 20, Corollary 20 and Remark 23 in [5], have versions that are valid for positive linear functionals and C-seminorms on unital Banach -algebras. However, the details are omitted. Acknowledgement 1. This work was done when the …rst author was at the Re- search Group in Mathematical Inequalities and Applications (RGMIA), Victoria University on sabbatical leave from Lorestan University. He thanks both institu- tions for their support.
References [1] F. F. BONSALL and J. DUNCAN, Complete Normed Algebras, Springer-Verlag, New York, 1973. [2] R.S DORAN and V.A. BELFI, Characterization of C-Algebras, Dekker, New York, 1986. [3] S. S. DRAGOMIR, A generalization of Grüss inequality in inner spaces and application,J. Math. Annal. Appl. 237(1999), 74-82. [4] S. S. DRAGOMIR, Some Grüss type inequalities in inner product spaces, J. Inequal. Pure Appl. Math. 4(2003), No 2, Article 42. [5] S. S. DRAGOMIR, Advances in Inequalities of the Schwarz, Grüss and Bessel Type in Inner Product Spaces, Nova Science puplishers Inc., New York, 2005. SCHWARZANDGRÜSSTYPEINEQUALITY 13
1 b [6] G. GRÜSS, Über das Maximum des absoluten Betrages von b a a f(x)g(x)dx 1 b f(x)dx b g(x)dx, Math. Z. 39(1934), 215-226. (b a)2 a a R ´ ´ [7] D. ILIŠEVIR C andR S. VAROŠANEC, Grüss type inequalities in inner product modules, Proc. Amer. Math. Soc. 133(11)(2005), 3271-3280. [8] G. J. MURPHY, C-Algebra and Operator Theory, Academic Press, 1990. [9] V. PTÁK, On the spectral radius in Banach algebras with involution, Bull. London Math. Soc., 2(1970), 327-334. [10] N. E. WEGGE-OLSEN, K-theory and C-algebras- A Friendly Approach, Oxford University Press, Oxford, 1993.
Department of Mathematics, Lorestan University, P.O.Box 68135-465, Khoramabad, Iran. E-mail address: [email protected]
School of Engineering, Science, Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia. E-mail address: [email protected]