Abstract
We disprove the Kadison-Singer property for the continuous subalgebra. We do this by making an explicit description of all pure states on this subalgebra (using a total set of states and the universal property of the Stone-Čech compactification) and some hard analysis arguments. As a consequence of the fact that the continuous subalgebra does not have the Kadison-Singer property, the mixed subalgebra does neither. This reduces the classification of the Kadison-Singer property to the question whether the discrete case has this property; this question was known as the Kadison-Singer conjecture.
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References
Wojtaszczyk, P.: A Mathematical Introduction to Wavelets. Cambridge University Press (1997)
Kadison, R.V., Singer, I.: Extensions of pure states. American Journal of Mathematics 81(2), 383–400 (1959)
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Stevens, M. (2016). The Continuous Subalgebra and the Kadison-Singer Conjecture. In: The Kadison-Singer Property. SpringerBriefs in Mathematical Physics, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-319-47702-2_7
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DOI: https://doi.org/10.1007/978-3-319-47702-2_7
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Publisher Name: Springer, Cham
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