MAPPING PRESERVING NUMERICAL RANGE OF OPERATOR PRODUCTS ON C*-ALGEBRAS

Abstract

Abstract. Let A and B be two unital C ∗ -algebras. Denote by W(a)the numerical range of an element a∈ A. We show that the conditionW(ax) = W(bx),∀x∈ A implies that a= b. Using this, among otherresults, it is proved that if φ : A → B is a surjective map such thatW(φ(a)φ(b)φ(c)) = W(abc) for all a,band c∈A, then φ(1) ∈Z(B) andthe map ψ= φ(1) 2 φis multiplicative. 1. IntroductionLet A be a C ∗ -algebra with unit 1 and let S(A) be the state space of A, i.e.,S(A) = {ϕ∈ A ′ : ϕ≥ 0,ϕ(1) = 1} (here A ′ is the topological dual of A). Foreach a∈ A, the algebraic numerical range V(a) and numerical radius v(a) aredefined byV(a) = {f(a) : f∈ S(A)} and v(a) = sup z∈V (a) |z|.By the Gelefand-Naimark theorem, every C ∗ -algebra may be viewed as a closed∗-subalgebra of B(H) where B(H) denotes the algebra of all bounded linearoperators acting on a Hilbert space H. It is well known that V(a) is the closureof W(a) and v(a) = w(a) = sup λ∈W(a) |λ|, where W(a) = {(at,t) : t∈ H,ktk = 1}and (,) denotes the inner product. Here W(a) is called the usual numericalrange of the operator a.In the last few decades, there has been a considerable interest in the problemof characterization of maps that preserves the numerical range or the numericalradius, see for instance the papers [4, 12, 13, 15] and the references therein.Notice that, based on the aforesaid, preserving the usual numerical range Wimplies the preservation of the spacial numerical range V. Therefore, we willconcentrate our study henceforth on W. Recently, Hou and Di described in [9]surjective maps on the algebra B(H) which preserves the numerical range of