dc.contributor.author Kaubya, Francis dc.date.accessioned 2021-05-12T09:40:44Z dc.date.available 2021-05-12T09:40:44Z dc.date.issued 2017 dc.identifier.uri http://hdl.handle.net/11071/11832 dc.description Paper presented at the 4th Strathmore International Mathematics Conference (SIMC 2017), 19 - 23 June 2017, Strathmore University, Nairobi, Kenya. en_US dc.description.abstract Let A denote a unital Banach algebra and SA denote its unit sphere. It was proved by F. F. Bonsall and J. Duncan that when the Banach algebra is unital, the algebra numerical range is identical to a subset of itself. Basing on this fact, we set out to investigate the relationship between the set of support functional for the unit ball (used construct the entire numerical range) and the normalized states (used to construct the identical subset of the numerical range). Indeed we have established that the union over all elements of SAof the sets of support functionals for the unit ball at each x ∈ SA i.e. Sx∈S D(A, x) is equal to the set ofe), when the Banach algebra is unital (ˆ is the unit element). The implicationis that even when the algebra is smooth, the set of normalized states i.e. D( , e) is not a singleton. Hence algebra numerical range is not singleton except when the element is a scalar mˆ le of the unit. Further, when defining smoothness for a unital Banach algebra, the unit element of the Banach algebra should beexcluded in the definition because the set of normalized states i.e. D( , e) may not be a singleton. Further, we established conditions under which statements P and P are equiv ˆ t;(P1) The union over all elements of SA of the sets of support functionals for the unit ball at each x ∈ SAi.e. Sx∈S D(A, x) is equal to the set of normalized states i.e. D(A, ecomponent algebra numerical range i.e.(P2) The algebra numerical range i.e. V (A, a) is equal to the ˆV (A, a, ˆ1) en_US dc.description.sponsorship Mbarara University of Science and Technology, Mbarara, Uganda en_US dc.language.iso en en_US dc.publisher Strathmore University en_US dc.subject Banach algebra en_US dc.subject Numerical range en_US dc.title On the Banach algebra numerical range en_US dc.type Article en_US
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### This item appears in the following Collection(s)

• SIMC 2017 
4th Strathmore International Mathematics Conference (June 19 – 23, 2017)