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**Inductive limits of interval algebras: The tracial state space.**
*(English)*
Zbl 0814.46050

An interval algebra is a \(C^*\)-algebra of the form \(A\otimes M_ n\) for some \(n\). The paper under review is the first of a series devoted to the study of AI-algebras, i.e. \(C^*\)-algebras which are \(*\)-isomorphic to inductive limits of sequences of interval algebras with unital connecting \(*\)-homomorphisms. The main results of the paper are contained in two theorems. The first one shows that for an AI-algebra \(A\) the following conditions are equivalent:

a) \(A\) is a UHF-algebra.

b) \(A\) has real rank zero.

c) \(A\) is the closed linear span of its projections.

d) \(A\) has only one trace state.

The second theorem shows that for every metrizable Choquet simplex \(S\) and every dense subgroup \(G\) of the rationals which contains the integers, there exists a sequence \((A_ n,\phi_ n)\) of interval algebras and injective connecting \(*\)-homomorphisms such that its inductive limit \(A\) is simple, \(K_ 0(A)\) is isomorphic to \(G\) and the tracial state space of \(A\) is affinely homeomorphic to \(S\).

In order to prove this interesting characterization the following result, due to the author and Jørgen Hoffmann-Jørgensen, is needed. Recall that a Markov operator is a unital positive linear map \(T: C(X)\to C(Y)\), where \(X\), \(Y\) are compact Hausdorff spaces. If \(X\) is pathconnected then the closed convex hull of the unital \(*\)-homomorphisms \(C(X)\to C(Y)\) is the strong operator topology coinciding with the set of Markov operators.

a) \(A\) is a UHF-algebra.

b) \(A\) has real rank zero.

c) \(A\) is the closed linear span of its projections.

d) \(A\) has only one trace state.

The second theorem shows that for every metrizable Choquet simplex \(S\) and every dense subgroup \(G\) of the rationals which contains the integers, there exists a sequence \((A_ n,\phi_ n)\) of interval algebras and injective connecting \(*\)-homomorphisms such that its inductive limit \(A\) is simple, \(K_ 0(A)\) is isomorphic to \(G\) and the tracial state space of \(A\) is affinely homeomorphic to \(S\).

In order to prove this interesting characterization the following result, due to the author and Jørgen Hoffmann-Jørgensen, is needed. Recall that a Markov operator is a unital positive linear map \(T: C(X)\to C(Y)\), where \(X\), \(Y\) are compact Hausdorff spaces. If \(X\) is pathconnected then the closed convex hull of the unital \(*\)-homomorphisms \(C(X)\to C(Y)\) is the strong operator topology coinciding with the set of Markov operators.

Reviewer: G.Corach (Buenos Aires)

### MSC:

46L05 | General theory of \(C^*\)-algebras |

47B38 | Linear operators on function spaces (general) |

46L30 | States of selfadjoint operator algebras |