On Numerical Ranges of Nilpotent Elements of C*-Algebra

Problem statement: Let A be a C*-algebra with unit 1. For each a ∈A, let V(a), ν(a) and ν0(a) denote its numerical range, numerical radius an d the distance from the origin to the boundary of its numerical range, respectively. Approach: If a is a nilpotent element of A with the power of nilpotency n, i.e., a n = 0, and ν(a) = (n-1) ν0(a). Results: We proved that V(a) = bW(A n), where b is a scalar and An is the strictly upper triangular n-by-n matrix wit h all entries above the main diagonal equal to one. Conclusion/Recommendations: We also completely determined the numerical range of such elements, by determining the numerical range o f W(An) and showed that the boundary of it does not contain any arc of circle.


INTRODUCTION
Let A be a C*-algebra with unit 1 and let S be the state space of A, i.e., S = {ϕ∈A*:ϕ≥0,ϕ(1) = 1}. For each a∈A, the C*-algebra numerical range V(a) and numerical radius ν(a) is defined, respectively, by: It is well known that V(a) is non empty, compact and convex subset of the complex plane, V(α1+βa) = α+βV(a) for a∈A and α,β∈ ℂ and if z∈V(a), z a ≤ [2] .
The notion of numerical range or the classical field of values was first introduced by Toeplitz in 1918 for matrices. This concept were independently extended by G. Lumer and F. Bauer in sixties to a bounded linear operator on an arbitrary Banach space. In 1975, Lightbourne and Martin [9] have extended this concept by employing a class of seminorms generated by a family of supplementary projections.
As an example, let A be the C*-algebra of all bounded linear operators on a complex Hilbert space H and T∈A. It is well known that V(T) is the closure of W(T), where: is the usual numerical range of the operator T. In this special case we denote the numerical radius of T and the distance from the origin to the boundary of its numerical range by w(T) and w 0 (T), respectively.
A complete survey on numerical range can be found in the books by Bonsall and Duncan [2,3] and the book by Gustafon and Rao [5] and we refer the reader to these books for general information and background.
In 1992, Haagerup and de la Harpe [6] have proved the following sharp estimate for the numerical radius of a nilpotent operator N: where, n is the power nilpotency of n. In 2004, Karaev give another proofs of the Haagerup-de la Harpe inequality.
In [4] the researchers have shown that if A is a nonzero nilpotent operator with the power of nilpotency n, with w(A)≤(n-1)w 0 (A) and if A attains its numerical radius then the following conditions are equivalent: • w(A) = (n-1)w 0 (A) • A is unitarily equivalent to an operator of the form ηA n ⊕A′, where η is a scalar satisfying |η| = 2w 0 (A) and A′ is some other operator:

MATERIALS AND METHODS
Let a be a nilpotent with the power of nilpotency n≥1, i.e., a n = 0 and ν(a) = (n-1)ν 0 (a), where ν 0 (a) denotes the distance from the origin to the boundary of its numerical range. In this study we show that V(a) = bW(A n ), where b is a scalar and by determining the boundary of numerical range of A n , we show that the ∂V(a) not inclusive a circle section. Actually, this study is an extension of the Haagerup-de and la Harpe inequality and the research of an earlier study by Gau [4] to the C*-algebra numerical range.

A short survey on W(A n ):
For the study of numerical ranges of finite matrices, the matrix-theoretic properties can be exploited to yield special tools which are not available for general operators. One important way to yield ∂W(A) is the Kippenhahn's result that the numerical range of A coincides with the convex hull of the real points of the dual curve of det(xReA + ylmA + zl) = 0 [4] . On the other hand, a parametric representation of the boundary W(A) can also be obtained from the largest eigenvalue of Re(e −iθ A) yielding useful information on W(A).

respectively.
As we mentioned, if A is a nilpotent operator on a Hilbert space H with nilpotency n that attaint its numerical radius and w(A) = (n-1)w 0 (A) then W(A) = bW(A n ), for some b. Hence for determining W(A) it is enough to compute W(A n ). The following theorem can be help us to find W(A).
By direct calculation we have: for -π≤θ≤π, θ ≠ 0 and λ′ (0) = 0 also: Now the proof is completed. This proof will help in discussing the following corollary: where, T a = π(a) [1] . Also a n = 0 implies that n a a T 0,T = is a nilpotent operator with nilpotency n. Then (i) follows from (1) and (2). Part (iv) follows from Theorem 1. Also w(T a ) = (n-1) w 0 (T a ) and so V(a) = W(T a ) = bW(A n ), where b is a scalar, which implies (ii). Finally (iii) follows from (i) and the facts that where, b is a scalar. Also ∂V(a) does not contain any arc of circle.
So, we can completely determine the numerical range of such elements.