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    On the Banach algebra numerical range
    (Strathmore University, 2017) Kaubya, Francis; Agure, John Ogonji; Emenyu, John
    Let A denote a unital Banach algebra and SA denote its unit sphere. It was proved by F. F. Bonsalland J. Duncan that when the Banach algebra is unital, the algebra numerical rangeis identical to a subset of itself. Basing on this fact, we have established that the union overall elements of SA of the sets of support functionals for the unit ball at each x ∈ SA i.e.Sx∈SD(A, x) is equal to the set of normalized states i.e. D(A, eˆwhen the Banach algebraAˆ).ˆeeeeA is unital (eˆ is the unit element). The implication is that even when the algebra is smooth, theset of normalized states i.e. D(A, e) is not a singleton. Hence algebra numerical range is notsingleton except when the element is a scalar multiple of the unit. Also, consider the statementsP1 and P2;(P1) The union over all elements of SA of the sets of support functionals for the unit ball ateach x ∈ SA i.e. Sx∈S D(A, x) is equal to the set of normalized states i.e. D(A, eˆ-component algebra numerical(P2) The algebra numerical range i.e. V (A, a) is equal to the eˆrange i.e. V (A, a, e).We have proved that the statements P1 and P2 are equivalent under a suitable condition. Further,F.F Bonsall and J. Duncan proved that for a unital Banach algebra, the numerical radius is anequivalent algebra norm. In that proof, the inequalities k > v(a) and k ≥ 1 "a" were used toconclude that 1 "a" ≤ v(a) (e is the irrational number 2.718...). However, these two inequalities lead to the undesired inequality 1 "a" ≥ v(a). In this dissertation we improve on this proof bymerging the property of the complex roots of unity used by Bonsall and Duncan together withthe geometric series argument and making a different choice of the arbitrary element b ∈ A ofnorm less than one to derive the desired inequality 1 "a" ≤ v(a).andˆD(A, eˆe) = 1}.) = {f ∈ SAt : f (ˆ Elements of D(A, x) are called support functionals for the unit ball at x ∈ SA while elements of D(A, e) are called normalized states. For each a ∈ A and x ∈ SA defineV (A, a, x) = {f (ax) : f ∈ D(A, x)}