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SUMMARY:Gerardo Escolano (University of Granada\, Spain)
DTSTART:20261111T200000Z
DTEND:20261111T210000Z
DTSTAMP:20261111T025541Z
UID:NYC-NCG/230
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NYC-NCG/230/
 ">The Mackey–Gleason–Bunce–Wright Theorem</a>\nby Gerardo Escolano (
 University of Granada\, Spain) as part of Noncommutative geometry in NYC\n
 \n\nAbstract\nThe classical Gleason theorem\, motivated by questions arisi
 ng in the mathematical foundations of quantum\nmechanics\, initiated the s
 tudy of measures on the lattice of projections of operator algebras. A maj
 or\ngeneralization of this result was obtained by Bunce and Wright\, who e
 stablished that every bounded finitely\nadditive Banach-space-valued measu
 re on the projections of a von Neumann algebra without type $I_2$ direct\n
 summands extends to a bounded linear operator on the whole algebra. Whethe
 r an analogous result holds for\narbitrary $JBW^∗$-algebras remained an 
 open problem for several decades.\nIn this talk\, we will present a recent
  solution to this long-standing problem [1]\, establishing a Mackey–\nGl
 eason–Bunce–Wright theorem for $JBW^∗$-algebras without type $I_2$ d
 irect summands. We will discuss some\nof the main ideas behind the proof\,
  with particular emphasis on the role played by the structure theory of\n$
 JBW^∗$-algebras and the Bunce–Wright equivalence of projections.\nFina
 lly\, we will explore the application of this theorem to the study of the 
 quasi-linearity problem for\nJordan–Banach algebras [2]\, including a to
 pological characterization of the linearity of quasi-linear functionals.\n
 \nReferences\n\n[1] G. M. Escolano\, A. M. Peralta and A. R. Villena\, The
  Mackey–Gleason–Bunce–Wright problem for vectorvalued\nmeasures on p
 rojections in a JBW∗-algebra\, Rev. Real Acad. Cienc. Exactas Fis. Nat. 
 Ser. A-Mat.\n120\, 113 (2026). https://doi.org/10.1007/s13398-026-01906-5.
 \n\n[2] G. M. Escolano\, The quasi-linearity problem for Jordan–Banach a
 lgebras: a topological characterization\,\naccepted in Journal of Operator
  Theory (2026). https://arxiv.org/abs/2605.06120.\n
LOCATION:https://researchseminars.org/talk/NYC-NCG/230/
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