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N. Christopher Phillips

Publications and source records attributed to N. Christopher Phillips.

At least 73 records · Page 4Linked to original sources
arXiv2004

Stable and real rank for crossed products by automorphisms with the tracial Rokhlin property

We introduce the tracial Rokhlin property for automorphisms of stably finite simple unital C*-algebras containing enough projections. This property is formally weaker than the various Rokhlin properties considered by Herman and Ocneanu, Kishimoto, and Izumi. Our main results are as follows. Consider a stably finite simple unital C*-algebra, and an automorphism of it which has the tracial Rokhlin property. Suppose the algebra has real rank zero and stable rank one, and suppose that the order on projections over the algebra is determined by traces. Then the crossed product by the automorphism also has these three properties. We also present examples of C*-algebras and automorphisms which satisfy the above assumptions, but such that crossed product algebras do not have tracial rank zero.

math.OA↗
arXiv2004

Furstenberg transformations on irrational rotation algebras

We introduce a general class of automorphisms of rotation algebras, the noncommutative Furstenberg transformations. We prove that fully irrational noncommutative Furstenberg transformations have the tracial Rokhlin property, which is a strong form of outerness. We conclude that crossed products by these automorphisms have stable rank one, real rank zero, and order on projections determined by traces (Blackadar's Second Fundamental Comparability Question). We also prove that several classes of simple quotients of the C*-algebras of discrete subgroups of five dimensional nilpotent Lie groups, considered by Milnes and Walters, are crossed products of simple C*-algebras (C*-algebras of minimal ordinary Furstenberg transformations) by automorphisms which have the tracial Rokhlin property. It follows that these algebras also have stable rank one, real rank zero, and order on projections determined by traces.

math.OA↗
arXiv2004

Crossed products by minimal homeomorphisms

Let X be an infinite compact metric space with finite covering dimension and let h be a minimal homeomorphism of X. Let A be the associated crossed product C*-algebra. We show that A has tracial rank zero whenever the image of K_0 (A) in the affine functions on the tracial state space of A is dense. As a consequence, we show that these crossed product C*-algebras are in fact simple AH algebras with real rank zero. When X is connected and h is further assumed to be uniquely ergodic, then the above happens if and only if the rotation number associated to h has irrational values. By applying the classification theorem for nuclear simple C*-algebras with tracial rank zero, we show that two such dynamical systems have isomorphic crossed products if and only if they have isomorphic scaled ordered K-theory.

math.OA↗
arXiv2004

Examples of different minimal diffeomorphisms giving the same C*-algebras

We give examples of minimal diffeomorphisms of compact connected manifolds which are not topologically orbit equivalent, but whose transformation group C*-algebras are isomorphic. The examples show that the following properties of a minimal diffeomorphism are not invariants of the transformation group C*-algebra: having topologically quasidiscrete spectrum; the action on singular cohomology (when the manifolds are diffeomorphic); the homotopy type of the manifold (when the manifolds have the same dimension); and the dimension of the manifold. These examples also give examples of nonconjugate isomorphic Cartan subalgebras, and of nonisomorphic Cartan subalgebras, of simple separable nuclear unital C*-algebras with tracial rank zero and satisfying the Universal Coefficient Theorem.

math.OA↗
arXiv2004

Real rank and property (SP) for direct limits of recursive subhomogeneous algebras

Let A be a unital simple direct limit of recursive subhomogeneous C*-algebras with no dimension growth. We give criteria which specify exactly when A has real rank zero, and exactly when A has the Property (SP): every nonzero hereditary subalgebra of A contains a nonzero projection. Specifically, A has real rank zero if and only if the natural map from K_0 (A) to the continuous affine functions on the tracial state space has dense range, A has the Property (SP) if and only if the range of this map contains strictly positive functions with arbitrarily small norm. By comparison with results for unital simple direct limit of homogeneous C*-algebras with no dimension growth, one might hope that weaker conditions might suffice. We give examples to show that several plausible weaker conditions do not suffice for the results above. If A has real rank zero and at most countably many extreme tracial states, we apply results of H. Lin to show that A has tracial rank zero and is classifiable.

math.OA↗
arXiv2003

Crossed products by finite cyclic group actions with the tracial Rokhlin property

We define the tracial Rokhlin property for actions of finite cyclic groups on stably finite simple unital C*-algebras. We prove that when the algebra is in addition simple and has tracial rank zero, then the crossed product again has tracial rank zero. Under a kind of weak approximate innerness assumption and one other technical condition, we prove that if the action has the the tracial Rokhlin property and the crossed product has tracial rank zero, then the original algebra has tracial rank zero. We give examples showing how the tracial Rokhlin property differs from the Rokhlin property of Izumi. We use these results, together with work of Elliott-Evans and Kishimoto, to give an inductive proof that every simple higher dimensional noncommutative torus is an AT algebra. We further prove that the crossed product of every simple higher dimensional noncommutative torus by the flip is an AF algebra, and that the crossed products of irrational rotation algebras by the standard actions of the cyclic groups of orders 3, 4, and 6 are simple AH algebras with real rank zero.

math.OA↗
arXiv2003

Symmetries of Kirchberg algebras

Let A be a separable unital nuclear purely infinite simple C*-algebra satisfying the Universal Coefficient Theorem, and such that the K_0-class of the identity is zero. We prove that every automorphism of order two of the K-theory of A is implemented by an automorphism of A of order two. As a consequence, we prove that every countable Z/2Z-graded module over the representation ring of Z/2Z is isomorphic to the equivariant K-theory for some action of Z/2Z on a separable unital nuclear purely infinite simple C*-algebra. Along the way, we prove that every not necessarily finitely generated module over the group ring of Z/2Z which is free as an abelian group has a direct sum decomposition with only three kinds of summands, namely the group ring itself and Z on which the nontrivial element of Z/2Z acts either trivially or by multiplication by -1.

math.OA↗
arXiv2002

A simple separable C*-algebra not isomorphic to its opposite algebra

We give an example of a simple separable C*-algebra which is not isomorphic to its opposite algebra. Our example is nonnuclear and stably finite, has real rank zero and stable rank one, and has a unique tracial state. It has trivial K_1, and its K_0-group is order isomorphic to a countable subgroup of the real numbers.

math.OA↗
arXiv2002

Crossed products of the Cantor set by free minimal actions of Z^d

Let d be a positive integer, let X be the Cantor set, and let Z^d act freely and minimally on X. We prove that the crossed product C* (Z^d, X) has stable rank one, real rank zero, and cancellation of projections, and that the order on K_0 (C* (Z^d, X)) is determined by traces. We obtain the same conclusion for the C*-algebras of various kinds of aperiodic tilings.

math.OA↗
arXiv2002

When are crossed products by minimal diffeomorphisms isomorphic?

This is a survey which discusses the isomorphism problem for both C* and smooth crossed products by minimal diffeomorphisms. For C* crossed products, examples demonstrate the failure of the obvious analog of the Giordano-Putnam-Skau Theorem on minimal homeomorphisms of the Cantor set. For smooth crossed products, there are many open problems.

math.OA↗
arXiv2001

Recursive subhomogeneous algebras

We introduce and characterize a particularly tractable class of unital type 1 C*-algebras with bounded dimension of irreducible representations. Algebras in this class are called recursive subhomogeneous algebras, and they have an inductive description (through iterated pullbacks) which allows one to carry over from algebras of the form C (X, M_n) many of the constructions relevant in the study of the stable rank and K-theory of simple direct limits of homogeneous C*-algebras. Our characterization implies in particular that if A is a separable C*-algebra whose irreducible representations all have dimension at most N (for some finite N), and if for each n the space of n-dimensional irreducible representations has finite covering dimension, then A is a recursive subhomogeneous algebra. We demonstrate the good properties of this class by proving subprojection and cancellation theorems in it.

math.OA↗
arXiv2001

Cancellation and stable rank for direct limits of recursive subhomogeneous algebras

We prove the following results for a unital simple direct limit $A$ of recursive subhomogeneous algebras with no dimension growth: (1) A has stable rank 1. (2) The projections in $M_{\infty} (A)$ satisfy cancellation: if $e \oplus q \sim f \oplus q$, then $e \sim f$. (3) $A$ satisfies Blackadar's Second Fundamental Comparability Question: if $p, q \in M_{\infty} (A)$ are projections such that $τ(p) < τ(q)$ for all normalized traces $τ$ on $A$, then $p$ is equivalent to a subprojection of $q$. (4) $K_0 (A)$ is unperforated for the strict order: if $η\in K_0 (A)$ and there is $n > 0$ such that $n η> 0$, then $η> 0$. The last three of these results hold under certain weaker dimension growth conditions and without assuming simplicity. We use these results to obtain previously unknown information on the ordered K-theory of the crossed product $C^* (Z, X, h)$ obtained from a minimal homeomorphism of an infinite finite dimensional compact metric space $X$. Specifically, $K_0 (C^* (Z, X, h))$ is unperforated for the strict order, and satisfies the following K-theoretic version of Blackadar's Second Fundamental Comparability Question: if $η\in K_0 (A)$ satisfies $τ_* (\et) > 0$ for all normalized traces $τ$ on $A$, then there is a projection $p \in M_{\infty} (A)$ such that $η= [p]$.

math.OA↗
arXiv1997

A Classification Theorem for Nuclear Purely Infinite Simple C*-Algebras

Starting from Kirchberg's theorems announced in 1994, namely O_2 tensor A is isomorphic to O_2 for separable unital nuclear simple A and O_infinity tensor A is isomorphic to A if in addition A is purely infinite, we prove that KK-equivalence implies isomorphism for nonunital separable nuclear purely infinite simple C*-algebras. It follows that if A and B are unital separable nuclear purely infinite simple C*-algebras which satisfy the Universal Coefficient Theorem, and if there is a graded isomorphism from K_* (A) to K_* (B) which preserves the class of the identity, then A is isomorphic to B. Our main technical results are, we believe, of independent interest. We say that two asymptotic morphisms t ---> ϕ_t and t ---> ψ_t from A to B are asymptotically unitarily equivalent if there exists a continuous unitary path t ---> u_t in the unitization B^+ such that || u_t ϕ_t (a) u_t^* - ψ_t (a) || ---> 0 for all a in A. Let A be separable, nuclear, unital, and simple, and let D be unital. We prove that any asymptotic morphism from A to K tensor O_infinity tensor D is asymptotically unitarily equivalent to a homomorphism, and two homotopic homomorphisms from A to K tensor O_infinity tensor D are asymptotically unitarily equivalent.

funct-an↗
arXiv1997

Embedding of exact C*-algebras and continuous fields in the Cuntz algebra O_2

We prove that any separable exact C*-algebra is isomorphic to a subalgebra of the Cuntz algebra ${\cal O}_2.$ We further prove that if $A$ is a simple separable unital nuclear C*-algebra, then ${\cal O}_2 \otimes A \cong {\cal O}_2,$ and if, in addition, $A$ is purely infinite, then ${\cal O}_{\infty} \otimes A \cong A.$ The embedding of exact C*-algebras in $\OA{2}$ is continuous in the following sense. If $A$ is a continuous field of C*-algebras over a compact manifold or finite CW complex $X$ with fiber $A (x)$ over $x \in X,$ such that the algebra of continuous sections of $A$ is separable and exact, then there is a family of injective homomorphisms $ϕ_x : A (x) \to {\cal O}_2$ such that for every continuous section $a$ of $A$ the function $x \mapsto ϕ_x (a (x))$ is continuous. Moreover, one can say something about the modulus of continuity of the functions $x \mapsto ϕ_x (a (x))$ in terms of the structure of the continuous field. In particular, we show that the continuous field $θ\mapsto A_θ$ of rotation algebras posesses unital embeddings $ϕ_θ$ in ${\cal O}_2$ such that the standard generators $u (θ)$ and $v (θ)$ are mapped to $\operatorname{Lip}^{1/2}$ functions.

funct-an↗
arXiv1995

Classification of direct limits of even Cuntz-circle algebras

We prove a classification theorem for purely infinte simple C*-algebras that is strong enough to show that the tensor products of two different irrational rotation algebras with the same even Cuntz algebra are isomorphic. In more detail, let C be be the class of simple C*-algebras A which are direct limits A = lim A_k, in which each A_k is a finite direct sum of algebras of the form C(X) \otimes M_n \otimes O_m, where m is even, O_m is the Cuntz algebra, X is either a point, a compact interval, or the circle S^1, and each map A_k ---> A is approximately absorbing. ("Approximately absorbing" is defined in Section 1 of the paper.) We show that two unital C*-algebras A and B in the class C are isomorphic if and only if (K_0 (A), [1_A], K_1 (A)) is isomorphic to (K_0 (B), [1_B], K_1 (B)). This class is large enough to exhaust all possible K-groups: if G_0 and G_1 are countable odd torsion groups and g is in G_0, then there is a C*-algebra A in C with (K_0 (A), [1_A], K_1 (A)) isomorphic to (G_0, g, G_1). The class C contains the tensor products of irrational rotation algebras with even Cuntz algebras. It is also closed under several natural operations.

funct-an↗
arXiv1995

C*-Algebras with the Approximate Positive Factorization Property

We say that a unital C*-algrebra A has the approximate positive factorization property (APFP) if every element of A is a norm limit of products of positive elements of A. (There is also a definition for the nonunital case.) T. Quinn has recently shown that a unital AF algebra has the APFP if and only if it has no finite dimensional quotients. This paper is a more systematic investigation of C*-algebras with the APFP. We prove various properties of such algebras. For example: They have connected invertible group, trivial K_1, and stable rank 1. In the unital case, the K_0 group separates the tracial states. The APFP passes to matrix algebras. and if I is an ideal in A such that I and A/I have the APFP, then so does A. We also give some new examples of C*-algebras with the APFP, including type II_1 factors and infinite-dimensional simple unital direct limits with slow dimension growth, real rank zero, and trivial K_1 group. An infinite- dimensional simple unital direct limit with slow dimension growth and with the APFP must have real rank zero, but we also give examples of unital algebras with the APFP which do not have real rank zero. Our analysis also leads to the introduction of a new concept of rank for a C*-algebra that may be of interest in the future.

funct-an↗
arXiv1995

Approximate Unitary Equivalence of Homomorphisms from O_infinity

We prove that if two nonzero homomorphisms from the Cuntz algebra O_infinity to a purely infinite simple C*-algebra have the same class in KK-theory, and if either both are unital or both are nonunital, then they are approximately unitarily equivalent. It follows that O_infinity is classifiable in the sense of Rordam. In particular, Rordam's classification theorem for direct limits of matrix algebras over even Cuntz algebras extends to direct limits involving both matrix algebras over even Cuntz algebras and corners of O_infinity for which the K_0 group can be an arbitrary countable abelian group with no even torsion.

funct-an↗