arXiv:1504.04569v1 [math.OA] 8 Apr 2015
A FORMULA FOR THE NUMERICAL RANGE OF ELEMENTARY OPERATORS ON C ∗ −ALGEBRA M. BARRAA
Abstract. Let A be a C ∗ −algebra with unit element 1 and unitary group U. Let a = (a1 , ..., ak ) and b = (b1 , ..., bk ) two k−tuples of elements in A. The elementary operator associated to a and b is defined P by Ra,b (x) = ki=1 ai xbi . In this paper we prove the following formula for the numerical range of Ra,b : k X V (Ra,b , B(A)) = [∪{V ( u∗ ai ubi , A) : u ∈ U }]− . i=1
This formulat solves the problem 4.5 of [6].
1. Introduction Let A be a complex unital Banach algebra. Let a = (a1 , ..., ak ) and b = (b1 , ..., bk ) two k−tuples of elements in A. The elementary operator associated to a and b is defined by: k X
Ra,b (x) =
ai xbi .
i=1
This is a bounded linear operator on A. We refer to [6, 4] for good survey of this class of operators. The numerical range of a ∈ A is defined by: V (a, A) = {f (a) :
f ∈ S},
where S is the set of states in A ( S = {f ∈ A∗ , kf k = 1 = f (e)}). We refer to [2, 3, 5] for the basic facts about numerical ranges. A fondamental example is the C ∗ −algebra B(H) of bounded linear operators on a complex Hilbert space H. In this case we write RA,B for the elementary operator defined by RA,B (X) =
k X
Ai XBi .
i=1
The numerical range of a generalized derivation in B(H) was studied by severals authors, see for instances [8, 15, 9]. The numerical range of an elementary operators was studied by Seddik [12, 14, 13]. In [6], L. Fialkow posed the following problem : Problem. Determine the numerical range and the essential numerical range of the elementary operator RA,B . In [1], we gave a formula for the numerical range of an elementary operator in the case of the operator algebra B(H). In this paper we extend this formula to the context of a C ∗ −algebra.
1991 Mathematics Subject Classification. 47A12, 47B47. Key words and phrases. Elementary operators, numerical range. 1
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M. BARRAA
2. An inclusion in Banach algebra An element u ∈ A is said to be unitary if u is invertible and kuk = ku−1 k = 1. Note that kuak = kauk = kak for any a ∈ A and any u unitary in A. The set of unitary elements is denoted by U (A) or simply U. Theorem 2.1. [16]. Let A be a complex Banach algebra with unit. Then for any a ∈ A we have \ V (a, A) = {λ : |λ − z| ≤ ka − z k}. z∈C I Note that V (u−1 au, A) = V (a, A) for any a ∈ A and any unitary u.
Proposition 2.1. Let A be a complex unital algebra. For a = (a1 , ..., ak ) and b = (b1 , ..., bk ) two k−tuples of elements in A, we have the following inclusion: k X u−1 ai ubi , A) : u ∈ U } ⊂ V (Ra,b , B(A)). ∪{V ( i=1
Proof. The norm of an elementary operator is defined by kRa,b k = Sup{kRa,b(x)k :
kxk ≤ 1}.
From theorem 2.1 we deduce V (Ra,b , B(A)) =
\
{λ :
|λ − z| ≤ kRa,b − zk}.
z∈C
And
k \ X u−1 ai ubi , A) = {λ : V( i=1
|λ − z| ≤ k
k X
u−1 ai ubi − zk}.
i=1
z∈C
which can be written k \ X u−1 ai ubi , A) = {λ : V( i=1
And hence
|λ − z| ≤ k
i=1
ai ubi − zuk}.
i=1
z∈C
k \ X u−1 ai ubi , A) = {λ : V(
k X
|λ − z| ≤ k(Ra,b − z)(u)k}.
z∈C
But k(Ra,b − z)(u)k ≤ kRa,b − zk, thus k X u−1 ai ubi , A) : u ∈ U } ⊂ V (Ra,b , B(A)). ∪{V ( i=1
If A is a unital C ∗ −algebra then u ∈ A is unitary if and only if u∗ u = uu∗ = e. Theorem 2.2. [11](Russo-Dy’s theorem). Let A be a C ∗ −algebra with unit element 1 and unitary group U. Then the closed unit ball in A is the closed convex hull of U. Corollary 2.1. Let a = (a1 , ..., ak ) and b = (b1 , ..., bk ) be two k-tuples of elements in A. Then kRa,b k = Sup{kRa,b(u)k :
u ∈ U (A)}
A FORMULA FOR THE NUMERICAL RANGE OF ELEMENTARY OPERATORS ON C ∗ −ALGEBRA
3
3. Main Result The main result of this paper is the following theorem : Theorem 3.1. Let A be a unital C ∗ −algebra. Let a = (a1 , ..., ak ) and b = (b1 , ..., bk ) two k−tuples of elements in A. Then k X u∗ ai ubi , A) : u ∈ U }]− . V (Ra,b , B(A)) = [∪{V ( i=1
Proof. We need only to show the inclusion ”⊂”. By theorem 2.1 kRa,b k = Sup{k
k X
u∗ ai ubi − zuk :
u ∈ U (A)}.
i=1
Hence, if λ ∈ V (Ra,b , B(A)), then, for all z ∈ C, λ ∈ {µ − z ≤ kRa,b − zk}. Let ǫ > 0, fixed, there exists a unitary uǫ ∈ A such that kRa,b − zk ≤ k
k X
u∗ǫ ai uǫ bi − zuk + ǫ.
i=1
Now using theorem 2.1, we get k \ X u−1 {λ : V( ǫ ai uǫ bi , A) = i=1
z∈C
|λ − z| ≤ k
k X
ai uǫ bi − zuǫ k}.
i=1
P So, there exists µ ∈ V ( ki=1 u−1 ǫ ai uǫ bi , A), such that kλ − µk ≤ ǫ. But ǫ is arbitrary, hence λ ∈ Pk −1 − [∪u∈U(A) V ( i=1 u ai ubi , A)] . 4. Some consequences 4.1. Generalized derivation. We can get many formulas by putting special elementary operators in our main theorem. For example, the generalized derivation defined by δa,b (x) = ax − xb, yields the following equality V (δa,b , B(A)) = [∪{V (uau∗ − b) : u ∈ U }]− . In the case of A = B(H) the algebra of bounded linear operators, it is well known [15], that V (δA,B , B(A)) = W (A)− − W (B)− . Which yields W (A)− − W (B)− = [∪{W (AU − U B) : U unitary }]− . An other case is the multiplication operator Ma,b defined by Ma,b (x) = axb. In [4], Harte asqued the following question: what is the numerical range of Mp,p ? where p is an orthogonal projection p = p∗ = p2 . From theorem 2.2 . we obtain: V (Ma,b , B(A)) = [∪{V (uau∗ b, A) : u ∈ U (A)]− . And so V (Mp,p , B(A)) = [∪{V (upu∗ p, A) : u ∈ U (A)]− . Hence Mp,p is hemitian if and only if upu∗ p is hermitian for all u ∈ U (A).
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M. BARRAA
4.2. Essential numerical range. Let E be a complex Banach space and B(E) the Banach algebra of all bounded linear operators acting on F. Denote by K(E) the ideal of all compact operators acting on E, and let π be the canonical projection from B(E) onto the Calkin algebra B(E)/K(E). Denote further by k.ke the essential norm kT ke = inf {kT + Kk : K ∈ K(E)}. The essential numerical range Ve (T ) of T is defined by Ve (T ) = V (π(T ), B(E)/K(E), k.ke ). Let A = (A1 , ..., Ak ) and B = (B1 , ..., Bk ) two k−tuples of elements in B(E). From Propsition 2.1, we get k X Ve ( (U −1 Ai U Bi )) ⊂ V (Rπ(A),π(B) , B(B(E)/K(E))). i=1
In the case of a Hilbert space H we obtain from theorem 3.1, that k X Ve ( (U ∗ Ai U Bi )) = V (Rπ(A),π(B) , B(B(H)/K(H))). (5) i=1
Numerical range of inner derivations δπ(T ) were studied by C.K.Fong [7], who proved : V (δπ(T ) ) = Ve (T ) − Ve (T ). Combining this formula with (5) yields V (δπ(T ) ) = Ve (T ) − Ve (T ) = Ve (U ∗ T U − T ). References [1] M. Barraa, A Formula for the Numerical Range of Elementary Operators, Hindawi Publishing Corporation, ISRN Mathematical Analysis, Volume 2014, Article ID246301. http://dxdoi.org/10.1155/2014/246301. [2] F.F Bonsall and J. Duncan, Numerical range vol I, Cambridge university press (1973). [3] F.F Bonsall and J. Duncan, Numerical range vol II, Cambridge university press (1973). [4] ,R.E. Curto and M. Mathieu Elementary operators and their Applications, 3nd international workshop 2009. [5] K.E. Gustafson and D.K.M. Rao, Numerical range, The field of values of linear operators and matrices, Springer(1997). [6] L. Fialkow, Elementary operators & Applications, Proceeding of the international workshop 1991. [7] C.K. Fong On the essential maximal numerical range, Acta Sci. Math., 41(1979), 307-315. [8] J. Kyle, Numerical ranges of derivations, Proc. Edinburgh. Math. Soc., 21(1978), 33-39. [9] K. Mattila Complex strict and uniformconvexity and hyponormal operators, Math. Proc. camb. Phil., 96(1984), 483-493. [10] A. Pietsch, Operator ideals, North-Holland Mathematical library vol.20 (1980). [11] B. Russo and H. Dye, A note on the unitary operators in C ∗ −algebras, Duke Math. J.33(1966), 413-416. J.Math. Anal.Appl. 33(1971), 212-219. [12] A. Seddik, The numerical range of elementary operators, Integ. equ.oper.theory 43(2002) pp. 248252. [13] A. Seddik, On the numerical range and norm of elementary operators, Linear and Multilinear Algebra., 2004, Vol. 52, Nos. 34, pp. 293302. [14] A. Seddik, The numerical range of elementary operators II, Linear Algebra and it’s Applications., 338(2001) 239-244. [15] S. Y. Shaw, On numerical ranges of generalized derivations and related properties, J. Aust. Math. Soc. Seri. A., 36(1984), 134-142. [16] J.G. Stampfli and J.P. Williams,Growth conditions and numerical range in a Banach algebra, Thoku Math J.,20(1968), 417-424. [17] R. M. Timoney, Norms of elementary operators, Ir. Math. Soc. Bull. 46(2001), 13-17. E-mail address:
[email protected] Department de Mathematics, Faculty of Sciences Semlalia, P.O. Box 2390, Marrakesh, Morocco.
