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SUMMARY:Vrej Zarikian (U.S. Naval Academy)
DTSTART:20210203T200000Z
DTEND:20210203T210000Z
DTSTAMP:20260420T052741Z
UID:NYC-NCG/36
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NYC-NCG/36/"
 >Unique Extension Properties for C*-Inclusions</a>\nby Vrej Zarikian (U.S.
  Naval Academy) as part of Noncommutative geometry in NYC\n\n\nAbstract\nL
 et $\\mathcal{A} \\subseteq \\mathcal{B}$ be a $C^*$-inclusion\, i.e.\, an
  inclusion of unital $C^*$-algebras with the same unit. Structural propert
 ies of the inclusion are often reflected by the fact that certain families
  of UCP (unital completely positive) maps on $\\mathcal{A}$ extend uniquel
 y to UCP maps on $\\mathcal{B}$. In particular\, depending on the structur
 e of $\\mathcal{A} \\subseteq \\mathcal{B}$\, it could be the case that\n\
 ni. every pure state on $\\mathcal{A}$ extends uniquely to a pure state on
  $\\mathcal{B}$ (i.e.\, $\\mathcal{A} \\subseteq \\mathcal{B}$ has the pur
 e extension property)\;\n\nii. a weak* dense set of pure states on $\\math
 cal{A}$ extend uniquely to pure states on $\\mathcal{B}$ (i.e.\, $\\mathca
 l{A} \\subseteq \\mathcal{B}$ has the almost extension property)\;\n\niii.
  the identity map $\\operatorname{id}:\\mathcal{A} \\to \\mathcal{A}$ exte
 nds uniquely to a UCP map $E:\\mathcal{B} \\to \\mathcal{A}$ (i.e.\, $\\ma
 thcal{A} \\subseteq \\mathcal{B}$ has a unique conditional expectation)\;\
 n\niv. the identity map $\\operatorname{id}:\\mathcal{A} \\to \\mathcal{A}
 $ extends uniquely to a UCP map $\\theta:\\mathcal{B} \\to I(\\mathcal{A})
 $\, where $I(\\mathcal{A})$ is the injective envelope of $\\mathcal{A}$ (i
 .e.\, $\\mathcal{A} \\subseteq \\mathcal{B}$ has a unique pseudo-expectati
 on).\n\nIn this talk\, we explore properties (i)-(iv) above\, with a speci
 al emphasis on abelian inclusions $C(X) \\subseteq C(Y)$ and inclusions $\
 \mathcal{A} \\subseteq \\mathcal{A} \\rtimes_r G$ arising from actions of 
 discrete groups. Applications to determining the simplicity of reduced cro
 ssed products are provided.\n
LOCATION:https://researchseminars.org/talk/NYC-NCG/36/
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