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SUMMARY:Lattices of Regular Ideals and Quotients - David Pitts (University
  of Nebraska)
DTSTART:20250715T111000Z
DTEND:20250715T113000Z
UID:TALK233053@talks.cam.ac.uk
DESCRIPTION:In this talk\, I will describe some results obtained jointly w
 ith J. Brown\, A. Fuller\, and S. Reznikoff.\n&nbsp\;\nLet $\\mathcal A$ b
 e a $C^*$-algebra and $X\\subseteq \\mathcal A$ and let $X^\\perp=\\{a\\in
  \\mathcal A: aX=Xa=\\{0\\}\\}$.\nAn ideal $J$ in a $C^*$-algebra $\\mathc
 al A$ is regular if $(J^\\perp)^\\perp=J$. Hamana has shown that the famil
 y of regular ideals in $\\mathcal A$ forms a Boolean algebra.\n&nbsp\;\nNo
 w let $(\\mathcal{A}\,\\mathcal{B})$ be a pair of $C\\sp *$-algebras with 
 $\\mathcal{B}\\subseteq\\mathcal{A}$ and assume $\\mathcal B$ contains an 
 approximate unit for $\\mathcal A$.\n&nbsp\;\nWhen the inclusion $(\\mathc
 al A\,\\mathcal B)$ has a faithful invariant pseudo-expectation\, I will d
 iscuss a description of the regular ideals of $\\mathcal A$ in terms of th
 e invariant regular ideals of $\\mathcal B$ and the pseudo-expectation.\n&
 nbsp\;\nI will also discuss the following results. If $(\\mathcal A\,\\mat
 hcal B)$ is a Cartan inclusion and $J$ is a regular ideal in $\\mathcal A$
 \, the quotient $(\\mathcal A/J\, \\mathcal B/(\\mathcal B\\cap J))$ is ag
 ain a Cartan inclusion. This result can be extended to the class of pseudo
 -Cartan inclusions.
LOCATION:External
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