SCHWARZ AND GRÜSS TYPE INEQUALITIES FOR C*-SEMINORMS AND POSITIVE LINEAR FUNCTIONALS ON BANACH -MODULES AMIR G. GHAZANFARI AND SEVER S. 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 sense1 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 SchwarzA 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.