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

TL;DR

This paper explores how the structure of commutative subalgebras of a unital C*-algebra encodes its finite-dimensionality and uniqueness, linking algebraic properties to order-theoretic features.

Contribution

It establishes that finite-dimensionality of a C*-algebra can be characterized by chain conditions on its poset of commutative subalgebras and shows order isomorphism implies *-isomorphism for finite-dimensional cases.

Findings

Finite-dimensional C*-algebras characterized by chain conditions on C(A)

Order isomorphism of C(A) implies *-isomorphism for finite-dimensional algebras

C(A) captures key algebraic properties of the original C*-algebra

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.