AF Algebras Research Articles

AF Algebras with Directed Sets of Finite Dimensional ∗ -Subalgebras

AF Algebras with Directed Sets of Finite Dimensional ∗ -Subalgebras

AF algebras with directed sets of finite-dimensional *-subalgebras

We characterize the unital A F AF algebras whose families of finite dimensional ∗ ^{\ast } -subalgebras are directed by inclusion. A representation theorem for the algebras of this class allows us to classify them up to ∗ ^{\ast } -isomorphisms.

Entropy of Automorphisms of AF Algebras

Here we consider a notion of topological entropy for automorphisms of AF algebras, based on a corresponding measure theoretic entropy of Connes and St0rmer [6] for automorphisms of hyperfinite von Neumann algebras with an invariant trace. We show how to compute the entropy of the shift on certain AF algebras associated with topological Markov chains. If jtf is a unital AF algebra, let T(jtf) denote the normalized traces on $/. If 4> E T(jtf), and B is a finite dimensional C*-subalgebra of $#, let E% denote the conditional expectation of stf onto B, relative to 4>. For simplicity, we will always only consider finite dimensional subalgebras, with the same unit as stf. If n e N9 let Sn denote the maps x from Z into j^+ with finite support such that £ x(al5..., an) = l. For l<Z<n, xESn, aeZ, put afeZ

AF algebras with a lattice of projections.

AF algebras with a lattice of projections.

Automorphisms of dimension groups and the construction of AF algebras

Automorphisms of dimension groups and the construction of AF algebras

Traces on simple AF C∗-algebras

The relations between the set of traces on a simple approximately finite dimensional C ∗-algebra A and the algebraic and geometric properties of the Elliott dimension group K 0( A) are studied. It is shown that every metrizable Choquet simplex occurs as the set of normalized traces of a simple unital AF algebra. A simple AF algebra can have both finite and infinite traces, so a finite simple AF algebra need not be algebraically simple. It is shown that a simple AF algebra is algebraically simple if and only if it has no infinite traces, and is stable if and only if it has no finite traces.

Approximately Finite Dimensional C ∗ -Algebras and Bratteli Diagrams

ABsTRAcr. We determine properties of an AF algebra by observing the characteristics of its diagram. In particular, we characterize AF algebras that are liminal, postliminal, antiliminal and with continuous trace; moreover, we characterize liminal AF algebras with Hausdorff spectrum. Some elementary examples of AF algebras with certain desired properties are constructed by using these characterizations.

A Simple C ∗ -Algebra with no Nontrivial Projections

A C*-algebra is constructed which is separable, simple, nuclear, nonunital, and contains no nonzero projections. Some results on automorphisms of AF algebras are also obtained. A C*-algebra is said to be projectionless if it contains no projections other than 1 (if present) and 0. It has long been an open question whether there exists a projectionless simple C*-algebra (see [13, p. 18], [6, 1.9.6], [8, p. 81], [14, p. 242]). In this paper we construct a projectionless simple separable nuclear nonunital C*-algebra. It is quite possible that the methods of this paper can be modified to yield a projectionless simple unital C*-algebra. It is conjectured that the C*-algebra generated by the regular representation of the free group on two generators (known to be simple and unital) is projectionless. 1. Outline of construction. The general method of construction is motivated by the construction of the Bunce-Deddens weighted shift algebras [5] as described by Green [12, p. 248]. The algebra A is constructed as an inductive limit of C*-algebra An, each of which is a continuous field algebra on a circle T with a constant simple fiber B with the embedding j,,: A, -->A,,+1 inducing the twice around map z -> z2of T onto T. The algebra B will be the (unique) simple unital AF algebra whose ordered group KO(B) is isomorphic to the additive group of real algebraic numbers [10, 2.2]. B has the following properties: (1) B has a unique normalized trace , which is faithful. (2) If p and q are projections in B, then p q if and only if (p) = T(q). (3) If X is any algebraic number with 0 B such that f(l) = a(f(O)). Prim(A(a)) = {J,: 0 e PROPOSITION 1.1. A(a) is projectionless if (and only if) a(1) #6 1. Received by the editors December 13, 1978. AMS (MOS) subject classifications (1970). Primary 46L05. ? 1980 American Mathematical Society 0002-9939/80/0000-01 60/$02.25

Approximately finite-dimensional 𝐶*-algebras and Bratteli diagrams

We determine properties of an AF algebra by observing the characteristics of its diagram. In particular, we characterize AF algebras that are liminal, postliminal, antiliminal and with continuous trace; moreover, we characterize liminal AF algebras with Hausdorff spectrum. Some elementary examples of AF algebras with certain desired properties are constructed by using these characterizations.

A simple 𝐶*-algebra with no nontrivial projections

A C ∗ {C^\ast } -algebra is constructed which is separable, simple, nuclear, non-unital, and contains no nonzero projections. Some results on automorphisms of AF algebras are also obtained.

Equilibrium states of a semi-quantum lattice system

As an intermediate situation between quantum and classical lattice systems we investigate a semi-quantum lattice system (such as an alloy of several kinds of atoms). The relevant observable algebra is a special type of AF algebras. We study equilibrium states of the system and obtain some equivalent equilibrium conditions.