Classifying Finite-Dimensional C*-Algebras by Posets of Their Commutative C*-Subalgebras

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. underlying topological vector space, but with multiplication defined by (a, b) → ba, where (a, b) → ab denotes the original multiplication. Since C(A) is always isomorphic to C(A op ) as poset, for each C*-algebra A, the existence of Connes' C*-algebra Ac shows that the order structure of C(A) is not always enough in order to reconstruct A. More recent counterexamples can be found in [37] and [38].Nevertheless, there are still problems one could study. For instance, in [10] Döring and Harding consider a functor similar to C, namely the functor V assigning to a von Neumann algebra M the poset V(M ) of its commutative von Neumann subalgebras, and prove that one can reconstruct the Jordan structure, i.e., the anticommutator (a, b) → ab + ba, of M from V(M ). Similarly, in [17], it is shown that if C(A) and C(B) are order isomorphic, then there exists a quasi-linear Jordan isomorphism between Asa and Bsa, the sets of self-adjoint elements of A and B, respectively. Here quasi-linear means linear with respect to elements that commute. In [18], it is even shown that this quasi-linear Jordan isomorphism is linear when A and B are AW*-algebras.Moreover, one could replace C(A) by a structure with stronger properties. An example of such a structure is an active lattice, defined in [25], where Heunen and Reyes also show that this structure is strong enough to determine AW*-algebras completely.

Domains of Commutative C*-Subalgebras

“…This article fits into a wider programme: its associated partial order determines a C*-algebra to a great extent [21], [22], and has therefore become popular as a substitute [23], [24], [25]. In the case of deterministic computing it can be axiomatized [26].…”

On Defining Aw*-Algebras and Rickart C*-Algebras

“…Recent work by Hamhalter [9], Heunen and others, see [12,15,18] investigate to what extent the abelian *-subalgebras of a C*-algebra determine its structure. Also a number of interesting new results on AW*-algebras have been discovered; for example Hamhalter [8]; Heunen and Reyes [13] and [14].…”

The Many Classical Faces of Quantum Structures

“…[100]). A C*-algebra A is finite-dimensional if and only if C(A) is Artinian, if and only if C(A) is Noetherian.…”

“…[100,101]). If A = n i=1 M ni (C), then the C*-subalgebras M ni (C) correspond to directly indecomposable partially ordered subsets C i of C(A), and furthermore n i is the length of a maximal chain in C i .…”