AF Algebras Research Articles

Isometry groups of inductive limits of metric spectral triples and Gromov–Hausdorff convergence

AbstractIn this paper, we study the groups of isometries and the set of bi‐Lipschitz automorphisms of spectral triples from a metric viewpoint, in the propinquity framework of Latrémolière. In particular, we prove that these groups and sets are compact in the automorphism group of the spectral triple ‐algebra with respect to the Monge–Kantorovich metric, which induces the topology of pointwise convergence. We then prove a necessary and sufficient condition for the convergence of the actions of various groups of isometries, in the sense of the covariant version of the Gromov–Hausdorff propinquity, a noncommutative analogue of the Gromov–Hausdorff distance, when working in the context of inductive limits of quantum compact metric spaces and metric spectral triples. We illustrate our work with examples including AF algebras and noncommutative solenoids.

Lifting Bratteli diagrams between Krajewski diagrams: Spectral triples, spectral actions, and AF algebras

In this paper, we present a framework to construct sequences of spectral triples on top of an inductive sequence defining an $AF$-algebra. One aim of this paper is to lift arrows of a Bratteli diagram to arrows between Krajewski diagrams. The spectral actions defining Non-commutative Gauge Field Theories associated to two spectral triples related by these arrows are compared (tensored by a commutative spectral triple to put us in the context of Almost Commutative manifolds). This paper is a follow up of a previous one in which this program was defined and physically illustrated in the framework of the derivation-based differential calculus, but the present paper focuses more on the mathematical structure without trying to study the physical implications.

AF 𝐶*-algebras from non-AF groupoids

We construct ample groupoids from certain categories of paths, and prove that their C ∗ C^* -algebras coincide with the continued fraction approximately finite dimensional (AF) algebras of Effros and Shen. The proof relies on recent classification results for simple nuclear C ∗ C^* -algebras. The groupoids are not principal. This provides examples of Cartan subalgebras in the continued fraction AF algebras that are isomorphic, but not conjugate, to the standard diagonal subalgebras.

Nuclear dimension of extensions of O∞-stable algebras

We obtain an improved upper bound for the nuclear dimension of extensions of O∞-stable C∗-algebras. In particular, we prove that the nuclear dimension of a full extension of an O∞-stable C∗-algebra by a stable AF algebra is one.

Representations of Dirichlet operator algebras

A Dirichlet operator algebra is a nonself-adjoint operator algebra $\mathcal{A}$ with the property that $\mathcal{A} + \mathcal{A}^*$ is norm-dense in the C$^*$-envelope of $\mathcal{A}.$ We show that, under certain restrictions, $\mathcal{A}$ has a family of completely contractive representations $\{\pi_i\}$ with the property that the invariant subspaces of $\pi_i(\mathcal{A})$ are totally ordered, and such that, for all $a \in \mathcal{A}, ||a|| = \sup_i ||\pi_i(a)||.$ The class of Dirichlet algebras includes strongly maximal triangular AF algebras, certain semicrossed product algebras, and gauge-invariant subalgebras of Cuntz C$^*$-algebras. The main tool is the duality theory for essentially principal etale groupoids.

Derivation-based noncommutative field theories on AF algebras

In this paper, we start the investigation of a new natural approach to “unifying” noncommutative gauge field theories (NCGFT) based on approximately finite-dimensional ([Formula: see text]) [Formula: see text]-algebras. The defining inductive sequence of an [Formula: see text] [Formula: see text]-algebra is lifted to enable the construction of a sequence of NCGFT of Yang–Mills–Higgs types. This paper focuses on derivation-based noncommutative field theories. A mathematical study of the ingredients involved in the construction of a NCGFT is given in the framework of [Formula: see text] [Formula: see text]-algebras: derivation-based differential calculus, modules, connections, metrics and Hodge *-operators, and Lagrangians. Some physical applications concerning mass spectra generated by Spontaneous Symmetry Breaking Mechanisms (SSBM) are proposed using numerical computations for specific situations.

Structure and $K$-theory of $\ell^p$ uniform Roe algebras

In this paper, we characterize when the \ell^p uniform Roe algebra of a metric space with bounded geometry is (stably) finite and when it is properly infinite in standard form for p\in [1,\infty) . Moreover, we show that the \ell^p uniform Roe algebra is a (non-sequential) spatial L^p AF algebra in the sense of Phillips and Viola if and only if the underlying metric space has asymptotic dimension zero. We also consider the ordered K_0 groups of \ell^p uniform Roe algebras for metric spaces with low asymptotic dimension, showing that (1) the ordered K_0 group is trivial when the metric space is non-amenable and has asymptotic dimension at most one, and (2) when the metric space is a countable locally finite group, the (ordered) K_0 group is a complete invariant for the (bijective) coarse equivalence class of the underlying locally finite group. It happens that in both cases the ordered K_0 group does not depend on p\in [1,\infty) .

The Possible Temperatures for Flows on a Simple AF Algebra

It is shown that for any infinite dimensional simple unital AF algebra A and any closed lower bounded set K of real numbers containing zero there is a flow on A for which the set of possible inverse temperatures is K.

Ground States for Generalized Gauge Actions on UHF Algebras

We describe the structure of ground states and ceiling states for generalized gauge actions on a UHF algebra. It is shown that both sets are affinely homeomorphic to the state space of a unital AF algebra, and that any pair of unital AF algebras can occur in this way, independently of the field of KMS states. In addition we study the subset of the ground states called $${\text {KMS}}_\infty $$ -states.

Classification of certain inductive limit actions of compact groups on AF algebras

Let [Formula: see text] be an AF algebra, [Formula: see text] be a compact group. We consider inductive limit actions of the form [Formula: see text], where [Formula: see text] is an action on the finite-dimensional C*-algebra [Formula: see text] which fixes each matrix summand. We give a complete classification up to conjugacy of such actions using twisted equivariant K-theory.

Classification of spatial Lp AF algebras

We define spatial [Formula: see text] AF algebras for [Formula: see text], and prove the following analog of the Elliott AF algebra classification theorem. If [Formula: see text] and [Formula: see text] are spatial [Formula: see text] AF algebras, then the following are equivalent: [Formula: see text] and [Formula: see text] have isomorphic scaled preordered [Formula: see text]-groups. [Formula: see text] as rings. [Formula: see text] (not necessarily isometrically) as Banach algebras. [Formula: see text] is isometrically isomorphic to [Formula: see text] as Banach algebras. [Formula: see text] is completely isometrically isomorphic to [Formula: see text] as matricial [Formula: see text] operator algebras. As background, we develop the theory of matricial [Formula: see text] operator algebras, and show that there is a unique way to make a spatial [Formula: see text] AF algebra into a matricial [Formula: see text] operator algebra. We also show that any countable scaled Riesz group can be realized as the scaled preordered [Formula: see text]-group of a spatial [Formula: see text] AF algebra.

Bratteli diagrams via the De Concini–Procesi theorem

An AF algebra [Formula: see text] is said to be an AF[Formula: see text] algebra if the Murray–von Neumann order of its projections is a lattice. Many, if not most, of the interesting classes of AF algebras existing in the literature are AF[Formula: see text] algebras. We construct an algorithm which, on input a finite presentation (by generators and relations) of the Elliott semigroup of an AF[Formula: see text] algebra [Formula: see text], generates a Bratteli diagram of [Formula: see text] We generalize this result to the case when [Formula: see text] has an infinite presentation with a decidable word problem, in the sense of the classical theory of recursive group presentations. Applications are given to a large class of AF algebras, including almost all AF algebras whose Bratteli diagram is explicitly described in the literature. The core of our main algorithms is a combinatorial-polyhedral version of the De Concini–Procesi theorem on the elimination of points of indeterminacy in toric varieties.

The Cuntz–Toeplitz algebras have nuclear dimension one

We prove that unital extensions of Kirchberg algebras by separable stable AF algebras have nuclear dimension one. The title follows.

C*-algebras with and without ≪-increasing approximate units

For elements a,b of a C*-algebra we denote a=ab by a≪b. We show that all ω1-unital C*-algebras have ≪-increasing approximate units, extending a classical result for σ-unital C*-algebras. We also construct (in ZFC) the first examples of C*-algebras with no ≪-increasing approximate unit. One of these examples is a C*-subalgebra of B(ℓ2(ω1)). These examples are, by necessity, not approximately finite dimensional (AF), but some of them are still scattered and so locally finite dimensional (LF) in the sense of Farah and Katsura. It follows that there are scattered C*-algebras which are not AF. We further show that the existence of a C*-subalgebra of B(ℓ2) with no ≪-increasing approximate unit or the existence of an LF but not AF C*-subalgebra of B(ℓ2) are independent of ZFC. Our examples also show that an extension of an AF algebra by an AF algebra does not have to be an AF algebra in the nonseparable case.

THE UNITAL EXT-GROUPS AND CLASSIFICATION OFC*-ALGEBRAS

AbstractThe semigroups of unital extensions of separableC*-algebras come in two flavours: a strong and a weak version. By the unital Ext-groups, we mean the groups of invertible elements in these semigroups. We use the unital Ext-groups to obtainK-theoretic classification of both unital and non-unital extensions ofC*-algebras, and in particular we obtain a completeK-theoretic classification of full extensions of UCT Kirchberg algebras by stable AF algebras.

Crossed Products of Operator Algebras

We study crossed products of arbitrary operator algebras by locally compact groups of completely isometric automorphisms. We develop an abstract theory that allows for generalizations of many of the fundamental results from the selfadjoint theory to our context. We complement our generic results with the detailed study of many important special cases. In particular we study crossed products of tensor algebras, triangular AF algebras and various associated C*-algebras. We make contributions to the study of C*-envelopes, semisimplicity, the semi-Dirichlet property, Takai duality and the Hao-Ng isomorphism problem. We also answer questions from the pertinent literature.

Approximate equivalence of representations of AF algebras into semifinite von Neumann algebras

Approximate equivalence of representations of AF algebras into semifinite von Neumann algebras

N-parameter Fibonacci AF-Algebras are Isomorphic to Their Transpose AF-Algebras

An n-parameter Fibonacci AF-algebra $$\mathcal A$$ is any stationary AF $$C^*$$ -algebra with incidence matrix K of the form: $$\begin{aligned} K=\left( \begin{array}{ccccc}m_1&{}\quad m_2&{}\quad \cdots &{}\quad m_{n-1}&{}\quad m_n\\ 1&{}\quad 0&{}\quad \cdots &{}\quad 0&{}\quad 0\\ 0&{}\quad 1&{}\quad \cdots &{}\quad 0&{}\quad 0\\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots &{}\quad \vdots \\ 0&{}\quad 0&{}\quad \cdots &{}\quad 1&{}\quad 0\end{array}\right) , \end{aligned}$$ where $$n\ge 2$$ and the $$m_i$$ are nonnegative integers with $$m_n\not =0$$ . We write $${\mathcal {A}}={\mathcal {A}}_K$$ . The main result of the present paper is that $${\mathcal {A}}_K$$ is isomorphic to its transpose AF-algebra, $${\mathcal {A}}_{K^t}$$ .

On a lifting question of Blackadar

Let A be the AF algebra whose scaled ordered group K0(A) is (G⊕H,(G+∖{0})⊕H∪{(0,0)},g˜⊕0), where (G,G+,g˜) is the scaled ordered group K0(B) of a unital simple AF algebra B, and H is a countable torsion-free Abelian group. Let σ be an order 2 scaled ordered automorphism of K0(A), defined by σ(g,h)=(g,−h), where (g,h)∈G⊕H. We show that there is an order 2 automorphism α of A such that α∗=σ. This gives a partial answer to a lifting question posed by Blackadar. Incidentally, the lift α we construct has the tracial Rokhlin property. Consequently, the crossed product C∗(Z2,A,α) is a unital simple AH algebra with no dimension growth.

Word problems in Elliott monoids

Algorithmic issues concerning Elliott local semigroups are seldom considered in the literature, although these combinatorial structures completely classify AF algebras. In general, the addition operation of an Elliott local semigroup is partial, but for every AF algebra B whose Murray–von Neumann order of projections is a lattice, this operation is uniquely extendible to the addition of an involutive monoid E(B). Let M1 be the Farey AF algebra introduced by the present author in 1988 and rediscovered by F. Boca in 2008. The freeness properties of the involutive monoid E(M1) yield a natural word problem for every AF algebra B with singly generated E(B), because B is automatically a quotient of M1. Given two formulas ϕ and ψ in the language of involutive monoids, the problem asks to decide whether ϕ and ψ code the same equivalence of projections of B. This mimics the classical definition of the word problem of a group presented by generators and relations. We show that the word problem of M1 is solvable in polynomial time, and so is the word problem of the Behncke–Leptin algebras An,k, and of the Effros–Shen algebras Fθ, for θ∈[0,1]∖Q a real algebraic number, or θ=1/e. We construct a quotient of M1 having a Gödel incomplete word problem, and show that no primitive quotient of M1 is Gödel incomplete.