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arXiv · 1504.02730

Domains of commutative C*-subalgebras

Chris Heunen·Bert Lindenhovius
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Abstract

A C*-algebra is determined to a great extent by the partial order of its commutative C*-algebras. We study order-theoretic properties of this dcpo. Many properties coincide: the dcpo is, equivalently, algebraic, continuous, meet-continuous, atomistic, quasi-algebraic, or quasi-continuous, if and only if the C*-algebra is scattered. For C*-algebras with enough projections, these properties are equivalent to finite-dimensionality. Approximately finite-dimensional elements of the dcpo correspond to Boolean subalgebras of the projections of the C*-algebra, which determine the projections up to isomorphism. Scattered C*-algebras are finite-dimensional if and only if their dcpo is Lawson-scattered. General C*-algebras are finite-dimensional if and only if their dcpo is order-scattered.

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Chris Heunen, Bert Lindenhovius. 2015-04-10. Domains of commutative C*-subalgebras. https://doi.org/10.1017/s0960129518000464

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