The Mackey–Gleason–Bunce–Wright Theorem

Abstract: The classical Gleason theorem, motivated by questions arising in the mathematical foundations of quantum mechanics, initiated the study of measures on the lattice of projections of operator algebras. A major generalization of this result was obtained by Bunce and Wright, who established that every bounded finitely additive Banach-space-valued measure on the projections of a von Neumann algebra without type $I_2$ direct summands extends to a bounded linear operator on the whole algebra. Whether an analogous result holds for arbitrary $JBW^∗$-algebras remained an open problem for several decades. In this talk, we will present a recent solution to this long-standing problem [1], establishing a Mackey– Gleason–Bunce–Wright theorem for $JBW^∗$-algebras without type $I_2$ direct summands. We will discuss some of the main ideas behind the proof, with particular emphasis on the role played by the structure theory of $JBW^∗$-algebras and the Bunce–Wright equivalence of projections. Finally, we will explore the application of this theorem to the study of the quasi-linearity problem for Jordan–Banach algebras [2], including a topological characterization of the linearity of quasi-linear functionals.

References

[1] G. M. Escolano, A. M. Peralta and A. R. Villena, The Mackey–Gleason–Bunce–Wright problem for vectorvalued measures on projections in a JBW∗-algebra, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 120, 113 (2026). doi.org/10.1007/s13398-026-01906-5.

[2] G. M. Escolano, The quasi-linearity problem for Jordan–Banach algebras: a topological characterization, accepted in Journal of Operator Theory (2026). arxiv.org/abs/2605.06120.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

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