A.K. Gaur. Department of Mathematics, Duquesne University Pittsburgh, Pennsylvania 15282,. U.S.A.. Mursaleen. Department of Mathematics, A.M.U. Aligarh, ...
KYUNGPOOK Math. J. 37(1997), 221-226
Cones in Banach Algebras A.K. Gaur Department of Mathematics, Duquesne University Pittsburgh, Pennsylvania 15282, U.S.A. Mursaleen Department of Mathematics, A.M.U. Aligarh, U.P. 202002, India (1991 AMS Classification Number : 46J99,46J15) In Banach algebra A, the set of all hermitian elements and the set of all positive elements are studied with respect to the numerical range of these elements in A. We extend certain results concerning the cone of positive elements in a unital Banach algebra to the case where the algebra has no identity (of norm 1), but its norm is regular. For normal elements of A, a characterization of the spectral state is also given.
1. Introduction Let A be a unital Banach algebra with ||1|| = 1. For each x ∈ A, the subset of the complex plane C, V (A, x) = {f (x) : f ∈ A′ , f(1) = 1 = ||f ||} is called the unital numerical range of x, where A′ is the dual space of A [1]. The algebra A is said to have regular norm if for each a ∈ A, sup||x||≤1 ||ax|| = ||a||. If A has an identity 1 of norm one or has an approximate identity, then it has the regular norm. (See Proposition 3.2 in [4].) An example of such an algebra is L1 (R). Also suppose that A+ is endowed with the usual norm ||(λ, x)|| = |λ| + ||x||, for all x in A. This A+ is called the unitization of A. For more applications of the regular norm see [4, 6, 8, 9]. In [3] the spatial numerical range of an element in A is defined as follows: VA (a) = {f(ax) : f ∈ DA (x), x ∈ S(A)} where DA (x) = {f ∈ A′ : ||f || = 1 = f (x)} and S(A) = {x ∈ A : ||x|| = 1}. One can also define the spatial numerical range as the union of all the relative numerical ranges. (For a detailed description see [4, 5 and 7].) That is, ◦
VA (a) = ∪||b||=1 {V b (A, a)}, (Received : 30 April 1996).
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where V b (A, a) is the relative numerical range of a relative to b. The nonnegative number vA (a) = {|λ| : λ ∈ VA (a)} is called the spatial numerical radius of a. In [3], Theorem 2.3, it is proved that if A is a Banach algebra without identity and if A has the regular norm, then for each a ∈ A, V (A+ , a) = co VA (a) where co(X) denotes the closure of the convex hull of a set X. Hence this theorem and other results of [3] are true if A has an approximate identity of norm less than or equal to one. An element a in A is called Hermitian (i.e., A ∈ H(A), where H(A) is the set of all Hermitian elements of A), if VA ⊆ R and is positive if VA ⊆ R+ . (See definition 3.1 in [3].) If A is a unital Banach algebra, then H(A) is a real Banach space. Moreover, if K(A) denotes the set of all positive elements of A, then K(A) is a generating cone in H(A) (since 1 ∈ A, being interior to K(A), is also an order unit). It is natural to ask the following question: If A has no identity, which of the above properties of H(A) and K(A) persist? In Section 2 below, we give an answer to the above question and we apply this result to a B ∗ -algebra with the regular norm. More applications of the regular norm and a characterization of spectral state are considered in Section 3. The main results of this note are as follows: (i) Relation between H(A) and K(A): K(A) is a closed, normal, and generating cone in H(A) (Theorem 2.1). (ii) Applications to B∗ -algebra: (iii) The set of all spectral states of A is characterized: A functional f on A is a spectral state if and only if f(a) ∈ co spA (a), for all normal elements of A, where spA (x) is the spectrum of x in A. (iv) An example of a Banach algebra A containing a nonhermitian element a, for which V (A, a) does not equal co(spA (a)). 2. Cones in the set of hermitian elements. In this section we concentrate on the relationship between H(A) and K(A). Theorem 2.1. Let A be a Banach algebra with the regular norm. Then K(A) is closed, normal, and generating cone in H(A). Proof. From the definition of positive elements and from Lemma 2.2 of [3] it follows that K(A) + K(A) ⊂ K(A) and R+ K(A) ⊆ K(A).
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First, consider the case when A has an identity 1 of norm one. Then K(A) is a closed and normal cone in H(A) with 1 in its interior [1]; hence, K(A) generates H(A). Next, we prove that K(A) is normal even if A has no identity. Let a, b ∈ K(A). Then we claim that (1) vA (a) ≤ vA (a + b). Since a, b ∈ K(A), we have VA (a) ⊆ R+ and VA (b) ⊆ R+ . For each c in algebra A and for each f ∈ DA (c), f (ac) ≥ 0, f (bc) ≥ 0 and f (ac) + f (bc) ≥ f (ac). Therefore, from (1) it follows that vA (a) ≤ sup{|µ| : µ ∈ Vc (A, a + b)} where µ = f (ac) + f (bc), which shows that vA (a) ≤ vA (a + b). Thus, K(A) is normal since vA is a monotone norm on A. It remains to prove that K(A) generates H(A) if A has the regular norm. Since K(A+ ) generates H(A+ ), for h ∈ H(A) we have (0, h) = (λ1 , h1 ) − (λ2 , h2 ), λ1 ≥ 0, λ2 ≥ 0, and h1 , h2 ∈ K(A). Hence H(A) = K(A) − K(A). This completes the proof. Definition 2.1. An element x of A is self-adjoint if x = x∗ . Corollary 2.1. Let A be a B∗ -algebra with the regular norm. Then every selfadjoint element of A is hermitian and H(A) = K(A) − K(A).
Proof. Let h ∈ A such that h∗ = h. Then by Theorem 2.1, [6], we have h ∈ H(A) and hence by Theorem 2.1, K(A) generates H(A). The following example shows that H(A) = K(A) −K(A) even if the given norm is not equivalent to the regular norm. Example 2.1. Consider A = l1 with pointwise multiplication. Take 1 1 1 a= , , . . . , , 0, . . . n n n which is terminating after n ≥ 2 entries, then ||a||1 = 1, but ||ax||1 ≤
1 ||x||1 . n
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and this can be arbitrarily small compared with ||x||1 . Hence the norm is not even equivalent to the regular norm. But it is easy to identify the Hermitian and positive elements (they are respectively the real and the positive sequences since the numerical range lies between the convex hull of the range and its closure), and H(A) = K(A) − K(A). Remark 2.1. (open question). It is still unknown how much we can relax the requirements of the regular norm to keep the positive cone generating the space of Hermitian elements. Lemma 2.1. Let A be a B ∗ -algebra with the regular norm. Then K(A) is a proper convex, closed, and normal cone in H(A). Proof. It is obvious to see that K(A) + K(A) ⊂ K(A) and αK(A) ⊂ K(A) for each positive real α. Now we claim that K(A) ∩ (−K(A)) = {0}, where 0 denotes the zero element in H(A). Let a ∈ K(A) ∩ (−K(A)). Then by Proposition 2.2 of [3], VA (a) ⊆ R+ and − VA (a) = VA (−a) ⊆ R+ That is VA (a) ⊆ R+ ∩ R− and hence VA (a) = {0}. Therefore by Corollary 2.4 of [3], a = 0 and hence K(A) ∩ (−K(A)) = {0} which proves that K(A) is proper convex. That K(A) is closed is automatic. 3. Application of the regular norm. In this section we present additional benefits of having the regular norm. Let spA (x) and ρA (x) be the spectrum and the spectral radius of x in A, respectively. Theorem 3.1. Let A be a Banach algebra with the regular norm. If ρA (x) is the spectral radius of x and if for all x ∈ A, vA (x) = ρA (x), then ||a|| ≤ 21 exp(1) · ρA (a), for each a ∈ A. Proof. Since V (A+ , a) = co VA (a), for all a ∈ A (Theorem 2.3 of [3]), we have ρA+ (a) = vA+ (a) = vA (a) = ρA (a), for all a ∈ A; hence theorem follows from Theorem 5, p. 49, [2]. Corollary 3.1. If vA (a) = 1 and P is a polynomial that maps the unit disc into itself, then 1 ||P (a)|| ≤ exp(1). 2
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Proof. Follows from the Theorem 3.1 Definition 3.1. An element a ∈ A is normal if a = h + ik with h, k ∈ H(A), and hk = kh. We call an f ∈ A′ a spectral state of A, if f (a) ∈ co(spA (a)) for every a ∈ A. We write SP (A) for the set of all spectral states of A. Theorem 3.2. Let A be a complex Banach algebra with the regular norm. Then for all normal elements a ∈ A, f ∈ SP (A) if and only if f (a) ∈ co spA (a). Proof. If f ∈ SP (A), then f (a) ∈ co spA (a) = co spA+ (a) for all a ∈ A. From the fact that V (A+ , a) = co(spA+ (a)), [1], it follows that f(a) ∈ V (A+ , a) for each a ∈ A and hence the theorem follows from the Theorem 2.3 of [3]. Now we give an example of an algebra and a nonhermitian element a in this algebra such that VA (a) = co spA (a). Example 3.1. Suppose B(C 2 ) = A is the algebra of all bounded liner operators on C 2 . (See also [6], Example 2.1.) 1 1 Let a = and the norm on A be defined by ||m||∞ = maxi=1,2 2j=1 |mij |. 1 1 Note that this is an operator if the norm on C 2 is l∞ -norm. Let us define the functional φ ∈ A′ by φ(x) = trace(φT x). The norm of φ is given by 2 |trace(φT x)| ||φ|| = sup = max (|φji |) j=1,2 ||x|| x=0 i=1 The state space of A is the following: φ11 + φ22 = 1; max(|φ11 |, |φ22 |) + max(|φ12 |, |φ22 |) = 1. Let 0 ≤ α ≤ 1, |β| ≤ 1 − α and |γ| ≤ α. Then α β φ= . γ 1−α Hence V (A, a) (unital numerical range) is given by V (A, a) = {1 + β + γ : |β| + |γ| ≤ 1} α γ 1 1 where 1 + β + γ = trace . β 1−α 1 1 If β = γ = 0, then V (A, a) contains 1 and if β = γ = 2i , then V (A, a) contains 1 + i and hence V (A, a) is not a subset of R which proves that a ∈ / H(A). iθ Also, for every θ ∈ [−π, π], if β = γ = e2 , then 1 + β + γ = 1 + eiθ . Since V (A, a) is convex, we have D(1 : 1) ⊆ V (A, a), where D(1 : 1) is the disc of the center 1 and radius 1. In fact, if λ = 1 + β + γ, then |λ − 1| = |β + γ| ≤ |β| + |γ| ≤ 1, which shows that V (A, a) = D(1 : 1).
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It is straightforward to see that spA (a) = {0, 2} ⊆ V (A, a), but V (A, a) = co(spA (a)).
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F.F. Bonsall and J. Duncan, Numerical Ranges of Operators on Normed Spaces and of Elements of Normal Algebra, London Math. Soc. Lecture Note Series 2(1971).
[2]
F.F. Bonsall and J. Duncan Numerical Ranges II, Cambridge University Press, (1973).
[3]
A.K. Gaur and T. Husain, Spatial Numerical Ranges of Elements of Banach Algebras, Internat. J. of Math. and Math. Sci., 12(4)(1989), 633-640.
[4]
A.K. Gaur and T. Husain, Relative Numerical Ranges, Math. Japonica 36(1991), 127-135.
[5]
A.K. Gaur, Invariance of relative Numerical Ranges in Banach Algebras, Indian J. of Mathematics, 33(3)(1991), 255-261.
[6]
A.K. Gaur and N.R. Nandakumar, Geometry of Star Algebras and Sequence Spaces, Advances in Sequence Spaces and Applications. To appear.
[7]
A.K. Gaur and Z.V. Kovarik, it A Disc Characterization of Relative Numerical Ranges in Banach Algebras, Simon Steven, A Q.J. of Pure and Applied Mathematics, 66(12)(1992), 79-88.
[8]
A.K. Gaur and Z.V. Kovarik, Norms on Unitization of Banach Algebras, Proceedings of the A.M.S. 117(1)(1993), 111-113.
[9]
A.K. Gaur and Z.V. Kovarik, Norms, States, and Numerical Ranges on Direct Sums, J. of Analysis 11(1991), 155-164.
