Approximately Finite Dimensional Research Articles

Homotopies of Constant Cuntz Class

Abstract Let $A$ be a unital simple separable exact C$^{\ast }$-algebra that is approximately divisible and of real rank zero. We prove that the set of positive elements in $A$ with a fixed Cuntz class is path connected. This result applies in particular to irrational rotation algebras and approximately finite-dimensional (AF) algebras.

Universal AF-algebras

We study the approximately finite-dimensional (AF) C⁎-algebras that appear as inductive limits of sequences of finite-dimensional C⁎-algebras and left-invertible embeddings. We show that there is such a separable AF-algebra AF which is a split-extension of any finite-dimensional C⁎-algebra and has the property that any separable AF-algebra is isomorphic to a quotient of AF. Equivalently, by Elliott's classification of separable AF-algebras, there are surjectively universal countable scaled (or with order-unit) dimension groups. This universality is a consequence of our result stating that AF is the Fraïssé limit of the category of all finite-dimensional C⁎-algebras and left-invertible embeddings.With the help of Fraïssé theory we describe the Bratteli diagram of AF and provide conditions characterizing it up to isomorphisms. AF belongs to a class of separable AF-algebras which are all Fraïssé limits of suitable categories of finite-dimensional C⁎-algebras, and resemble C(2N) in many senses. For instance, they have no minimal projections, tensorially absorb C(2N) (i.e. they are C(2N)-stable) and satisfy similar homogeneity and universality properties as the Cantor set.

INDUCTIVE LIMITS OF C*-ALGEBRAS AND COMPACT QUANTUM METRIC SPACES

AbstractGiven a unital inductive limit of C*-algebras for which each C*-algebra of the inductive sequence comes equipped with a Rieffel compact quantum metric, we produce sufficient conditions to build a compact quantum metric on the inductive limit from the quantum metrics on the inductive sequence by utilizing the completeness of the dual Gromov–Hausdorff propinquity of Latrémolière on compact quantum metric spaces. This allows us to place new quantum metrics on all unital approximately finite-dimensional (AF) algebras that extend our previous work with Latrémolière on unital AF algebras with faithful tracial state. As a consequence, we produce a continuous image of the entire Fell topology on the ideal space of any unital AF algebra in the dual Gromov–Hausdorff propinquity topology.

A classification of flows on AFD factors with faithful Connes–Takesaki modules

We completely classify flows on approximately finite dimensional (AFD) factors with faithful Connes–Takesaki modules up to cocycle conjugacy. This is a generalization of the uniqueness of the trace-scaling flow on the AFD factor of type I I ∞ \mathrm {II}_\infty , which is equivalent to the uniqueness of the AFD factor of type I I I 1 \mathrm {III}_1 . In order to achieve this, we show that a flow on any AFD factor with faithful Connes–Takesaki module has the Rohlin property, which is a kind of outerness for flows introduced by Kishimoto and Kawamuro.

Classification of actions of compact abelian groups on subfactors with index less than 4

We classify actions of discrete abelian groups on some inclusions of von Neumann algebras, up to cocycle conjugacy. As an application, we classify actions of compact abelian groups on the inclusions of approximately finite dimensional (AFD) factors of type II1 with index less than 4, up to stable conjugacy.

Some model-theoretic correspondences between dimension groups and AF algebras

If A ⊆ B are structures for a first-order language S , A is said to be algebraically (existentially) closed in B just in case every positive existential (existential) S A -sentence true in B A is true in A A . In 1976 Elliott showed that unital AF (‘approximately finite-dimensional’) algebras are classified up to isomorphism by corresponding dimension groups with order unit. This paper shows that one dimension group with order unit is algebraically (existentially) closed in another just in case the corresponding AF algebras, viewed as metric structures, fall in the same relation.

The Radon–Nikodym problem for approximately proper equivalence relations

We study the Radon–Nikodym problem for approximately proper equivalence relations and more specifically the uniqueness of certain Gibbs states. One of our tools is a variant of the dimension group introduced in the study of approximately finite-dimensional (AF) algebras. As applications, we retrieve sufficient conditions for the uniqueness of traces on AF algebras and parts of the Perron–Frobenius–Ruelle theorem.

COCYCLE CONJUGACY OF ONE PARAMETER AUTOMORPHISM GROUPS OF AFD FACTORS OF TYPE III

We classify, up to cocycle conjugacy, one-parameter automorphism groups on an approximately finite dimensional (AFD) factor ℳ of type III with trivial Connes spectrum. Our goal is to find the complete cocycle conjugacy invariants for one-parameter automorphism groups on ℳ. We also study the relations between the flow of weights of ℳ and that of the crossed product ℳ ⋊α ℝ of ℳ by a one-parameter automorphism group α with Γ(α) = {0}. Moreover, we also study model realizations. "Model realizations" means that given certain commutative data, they can be realized as the complete cocycle conjugacy invariants of centrally free and centrally ergodic one-parameter automorphism groups on some properly infinite AFD von Neumann algebras.

A C*-Dynamical Entropy and Applications to Canonical Endomorphisms

For an automorphism α of a unital C*-algebra A, we give a definition of an entropy htφ(α) with respect to an α-invariant state φ of A. For Connes– Narnhofer–Thirring entropy hφ(α) and Voiculescu's topological entropy ht(α), in general hφ(α)⩽htφ(α)⩽ht(α), but the equalities do not always hold. We compute entropies of an endomorphism ρ with respect to the state ϕ defined from a left inverse of ρ. Cuntz's canonical inner endomorphism Φ of On satisfies hϕ(Φ)=htϕ(Φ), which is determined by the mean entropy of ϕ on the UHF (uniformly hyperfinite) algebra. If γ is Longo's canonical endomorphism for an irreducible graded standard AFD (approximately finite dimensional) inclusion N⊂M of infinite factors with finite index, then hϕ(γ)=(1/2)log(IndEγ), for the conditional expectation Eγ on γ(M).

Locally representable one-parameter automorphism groups of AF algebras and KMS states

As a mathematical model of classical statistical lattice systems we consider locally representable one-parameter automorphism groups of AF (approximately finite-dimensional) C ∗- algebras . We demonstrate how to construct examples which exhibit various types of non uniqueness of equilibrium states (or KMS states). Towards the classification problem of cocycle-conjugacy classes of one-parameter automorphism groups in general we introduce a new invariant called marginal spectrum (associated with ground states) and apply it to a special case of infinite tensor product type actions.

Outer Conjugacy Classes of Trace Scaling Automorphisms of Stable UHF Algebras

Connes [C] classified automorphisms of the II1 approximately finite-dimensional (AFD) factor R0;1 up to outer conjugacy. An automorphism of R0;1 determines a modulus or scaling factor on a trace ˆ . It follows from this classification and [CT] that automorphisms which do not leave a trace invariant are classified up to conjugacy by their modulus; for each 2 0; 1† there is up to conjugacy a unique automorphism of R0;1 which scales the trace by . Consequently, for such there is a unique factor of type III the Powers factor R whose associated type II1 factor is R0;1; necessarily, R ˆ R0;1 Z. A key idea for model building in Connes's work on the classification of automorphisms of AFD factors [C] and in subsequent work of Jones [J], Ocneanu [O], and Kawahigashi, Sutherland and Takesaki [KST] on amenable group actions is a non-commutative Rohlin lemma for an aperiodic automorphism of a finite von Neumann algebra. Recall that an automorphism of a von Neumann algebra M is said to be properly outer if the restriction to Me is outer for each non-zero invariant projection e, and aperiodic if every non-zero power is properly outer. Connes showed that an automorphism of M is properly outer if and only if for each non-zero projection e in M, and > 0, there exists a non-zero projection f in M such that kf f †k 0, there exist projections F1; ;Fn in M such that Pn jˆ1 Fj ˆ 1 and Fj† y Fj‡1 2 , j ˆ 1; . . . ; n, where Fn‡1 ˆ F1 and kxk2 ˆ x x† 1 2. MATH. SCAND. 83 (1998), 74^86

Uniformly T 2 Algebras in Approximately Finite-Dimensional C *-Algebras

There is extensive literature concerning approximately finite-dimensional (AF) C*-algebras. For example see [1, 5]. In recent years study has focused on non-self-adjoint subalgebras of AF C*-algebras ([4, 8, 9], for example) and this paper continues that theme. We define T(m, n) to be the block upper triangular subalgebra of Mm+n with self-adjoint part equal to Mm ⊕ Mn. The class of algebras that are considered here are norm closures of ascending chains of algebras of the form T(m, n) and are referred to as uniformly T2 algebras. This class of algebras properly contains the matroid C*-algebras of Dixmier.

Ulam Games, Łukasiewicz Logic, and AF C*-Algebras

Ulam asked what is the minimum number of yes-no questions necessary to find an unknown number in the search space (1, …, 2n), if up to l of the answers may be erroneous. The solutions to this problem provide optimal adaptive l error correcting codes. Traditional, nonadaptive l error correcting codes correspond to the particular case when all questions are formulated before all answers. We show that answers in Ulam’s game obey the (l+2)-valued logic of Łukasiewicz. Since approximately finite-dimensional (AF) C*-algebras can be interpreted in the infinite-valued sentential calculus, we discuss the relationship between game-theoretic notions and their C*-algebraic counterparts. We describe the correspondence between continuous trace AF C*-algebras, and Ulam games with separable Boolean search space S. whose questions are the clopen subspaces of S. We also show that these games correspond to finite products of countable Post MV algebras, as well as to countable lattice-ordered Specker groups with strong unit.

On the classification of the ordered groups associated with the approximately finite dimensional C∗-algebras

On the classification of the ordered groups associated with the approximately finite dimensional C∗-algebras