Abnormalities of an operator space

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dc.contributor.authorKrisko, Colin
dc.contributor.examiningcommitteeMartin, Robert (Mathematics)
dc.contributor.examiningcommitteeZhang, Yong (Mathematics)
dc.contributor.supervisorClouâtre, Raphaël
dc.date.accessioned2024-09-06T19:04:07Z
dc.date.available2024-09-06T19:04:07Z
dc.date.issued2024-08-24
dc.date.submitted2024-08-25T00:07:35Zen_US
dc.date.submitted2024-09-06T15:55:50Zen_US
dc.degree.disciplineMathematics
dc.degree.levelMaster of Science (M.Sc.)
dc.description.abstractA unital operator space 𝑋 is simply a subspace inside some C∗-algebra 𝒜 that contains the identity of 𝒜. Given a unital operator space 𝑋, a natural object of interest is the C∗-algebra generated by 𝑋. Notably, the structure of C∗(𝑋) is not determined by 𝑋 alone [Arv69]. To resolve this problem, one could instead consider the “smallest” C∗-algebra that can be generated by 𝑋. The C∗-envelope of 𝑋 is (if it exists) the C∗-algebra generated by a copy of 𝑋, that satisfies a particular universal property [Pau02]. An intimately related object is the Shilov ideal for 𝑋 which is (if it exists) the boundary ideal that contains every other boundary ideal [Arv69]. Here an ideal is called a boundary ideal when its quotient mapping restricts to a complete isometry on 𝑋 [Arv69]. The existence of these objects has attracted the attention of many: Notable contributions have been made by: Arveson [Arv69] [Arv08], Hamana [Ham79], Dritschel and McCullough [DM05], and Davidson and Kennedy [DK15]. More recently, Kakariadis [Kak13] presented a new proof of the existence. This proof yields a new characterization of the Shilov ideal as the restricted abnormalities of 𝑋. This thesis is devoted to the examination of the unrestricted abnormalities and studying what they can tell us about the operator space.
dc.description.noteOctober 2024
dc.identifier.urihttp://hdl.handle.net/1993/38538
dc.language.isoeng
dc.rightsopen accessen_US
dc.subjectMathematics
dc.subjectOperator Spaces
dc.subjectOperator Systems
dc.subjectC∗-algebra
dc.subjectC∗-envelope
dc.subjectShilov Ideal
dc.subjectBoundary Ideal
dc.subjectUnital Complete Contraction
dc.subjectUnital Completely Positive
dc.subjectInjective Cover
dc.subjectInjective Envelope
dc.titleAbnormalities of an operator space
dc.typemaster thesisen_US
local.subject.manitobano
project.funder.identifierhttps://doi.org/10.13039/100010318
project.funder.nameUniversity of Manitoba
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