Classifying finite-dimensional C*-algebras by posets of their commutative C*-subalgebras

Bert Lindenhovius

doi:10.1007/s10773-015-2817-6

Classifying finite-dimensional C-algebras by posets of their commutative C-subalgebras

Operator Algebras 2016-10-07 v3 Mathematical Physics math.MP

Authors: Bert Lindenhovius

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Abstract

We consider the functor C that to a unital C*-algebra A assigns the partial order set C(A) of its commutative C*-subalgebras ordered by inclusion. We investigate how some C*-algebraic properties translate under the action of C to order-theoretical properties. In particular, we show that A is finite dimensional if and only C(A) satisfies certain chain conditions. We eventually show that if A and B are C*-algebras such that A is finite dimensional and C(A) and C(B) are order isomorphic, then A and B must be *-isomorphic.

Keywords

finite-dimensional c*-algebras commutative c*-subalgebras partial order chain conditions order isomorphism star-isomorphism

Cite

@article{arxiv.1501.03030,
  title  = {Classifying finite-dimensional C*-algebras by posets of their commutative C*-subalgebras},
  author = {Bert Lindenhovius},
  journal= {arXiv preprint arXiv:1501.03030},
  year   = {2016}
}

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20 pages