Searcharxiv⌕ Search

arXiv subjects

N. Christopher Phillips

Publications and source records attributed to N. Christopher Phillips.

At least 19 recordsLinked to original sources
arXiv2026

Obstructions to homomorphisms between homogeneous C*-algebras

We develop a method to find new obstructions to the existence of homomorphisms between homogeneous C*-algebras with prescribed behavior on K-theory. As an application, we give a complete answer to a problem posed by Blackadar in 1993 concerning the existence of unital homomorphisms between algebras of matrix-valued functions on even spheres which are injective on K_0: we show that if n, d, k, r are natural numbers and k is not in the range n, n+1, ..., n+d-1, then there is no unital homomorphism from C (S^{2n}, M_r) to C (S^{2k}, M_{dr}) which is injective on K_0.

math.OA↗
arXiv2026

Pureness of Certain Crossed Product C*-Algebras

We establish comparison and divisibility properties for crossed product C*-algebras arising from automorphisms of algebras C (X, D) which lie over minimal homeomorphisms, from actions of compact groups which have finite Rokhlin dimension with commuting towers, and from actions of compact groups which have the restricted tracial Rokhlin property with comparison. We deduce that these crossed products we consider are pure, and conclude they have stable rank one, and in certain cases have real rank zero. We give examples in which these properties do not follow from previous results, in the case of C (X, D) due to the lack of Z-stability of D, the underlying topological spaces not being finite dimensional, or both.

math.OA↗
arXiv2026

An isomorphism theorem for infinite reduced free products

Let C_1, C_2, ... be a sequence of separable unital C*-algebras, equipped with faithful tracial states and satisfying a mild condition. Let A be a unital direct limit of one dimensional NCCW complexes, also equipped with a faithful tracial state. Suppose there is a unital trace preserving embedding of A in the Jiang-Su algebra which is an isomorphism on K-theory. (For example, A could be C([0,1]) with Lebesgue measure, or the Jiang-Su algebra itself.) Let D be the infinite reduced free product of the algebras C_n. Then the reduced free product A*D is isomorphic to D. If D is exact and the factors satisfy a blockwise real rank zero condition, then in place of A we can use C(X) for any contractible compact metric space X and any faithful tracial state on C(X). An example consequence is that the reduced free product of infinitely many copies of C([0,1]), with Lebesgue measure, is isomorphic to the reduced free product of infinitely many copies of the Jiang-Su algebra.

math.OA↗
arXiv2025

Compact Group Actions with the Tracial Rokhlin Property II: Examples and Nonexistence Theorems

In a previous paper, we introduced the restricted tracial Rokhlin property with comparison, a ``tracial'' analog of the Rokhlin property for actions of second countable compact groups on infinite dimensional simple separable unital C*-algebras. In this paper, we give three classes of examples of actions of compact groups which have this property but do not have the Rokhlin property, or even finite Rokhlin dimension with commuting towers. One class consists of infinite tensor products of finite group actions with the tracial Rokhlin property, giving actions of the product of the groups involved. The second class consists of actions of the circle group on simple unital AT~algebras. The construction of the third class starts with an action of the circle on the Cuntz algebra ${\mathcal{O}}_{\infty}$ which has the restricted tracial Rokhlin property with comparison; by contrast, it is known that there is no action of this group on ${\mathcal{O}}_{\infty}$ which has finite Rokhlin dimension with commuting towers. We can then tensor this action with the trivial action on any unital purely infinite simple separable nuclear C*-algebra. One also gets such actions on certain purely infinite simple separable nuclear C*-algebras by tensoring the AT~examples with the trivial action on ${\mathcal{O}}_{\infty}$; these are different. We also discuss other tracial Rokhlin properties for actions of compact groups, and prove that there is no direct limit action of the circle group on a simple AF~algebra which even has the weakest of these properties.

math.OA↗
arXiv2024

The K-Theory of a Simple Separable Exact C*-Algebra Not Isomorphic to Its Opposite Algebra

We construct uncountably many mutually nonisomorphic simple separable stably finite unital exact C$^\ast$-algebras which are not isomorphic to their opposite algebras. In particular, we prove that there are uncountably many possibilities for the $K_0$-group, the $K_1$-group, and the tracial state space of such an algebra. We show that these C*-algebras satisfy the Universal Coefficient Theorem. This is new even for the already known example of an exact C*-algebra nonisomorphic to its opposite algebra produced in earlier work.

math.OA↗
arXiv2023

Minimal dynamical systems on prime C*-algebras

We give a number of examples of exotic actions of locally compact groups on separable nuclear C*-algebras. In particular, we give examples of the following: (1) Minimal effective actions of ${\mathbb{Z}}$ and $F_n$ on unital nonsimple prime AF algebras. (2) For any second countable noncompact locally compact group, a minimal effective action on a separable nuclear nonsimple prime C*-algebra. (3) For any amenable second countable noncompact locally compact group, a minimal effective action on a separable nuclear nonsimple prime C*-algebra (unital when the group is ${\mathbb{Z}}$ or ${\mathbb{R}}$) such that the crossed product is $K \otimes {\mathcal{O}}_2$ (${\mathcal{O}}_2$ when the group is ${\mathbb{Z}}$). (4) For any second countable locally compact abelian group which is not discrete, an action on $K \otimes {\mathcal{O}}_2$ such that the crossed product is a nonsimple prime C*-algebra. In most of these situations, we can specify the primitive ideal space of the C*-algebra (of the crossed product in the last item) within a class of spaces.

math.OA↗
arXiv2023

The radius of comparison of $C (X)$

Let X be a compact Hausdorff space. Then the radius of comparison rc ( C (X)) is related to the covering dimension dim (X) by rc ( C (X)) \geq [ dim (X) - 7 ] / 2. Except for the additive constant, this improves a result of Elliott and Niu, who proved that if X is metrizable then rc (C (X)) \geq [ dim_{\mathbb{Q}} (X) - 4 ] / 2. There are compact metric spaces X for which the estimate of Elliott and Niu gives no information, but for which rc ( C (X)) is infinite or has arbitrarily large finite values.

math.OA↗
arXiv2023

Values of Rokhlin dimension for actions of compact groups

We show that any finite group admits actions on simple AF algebras with unique trace which have arbitrarily large finite values of Rokhlin dimension with commuting towers. We show similar results for actions of compact Lie groups, with AH algebras with no dimension growth in place of AF algebras. We also relate Rokhlin dimension to the G-index for actions of compact Lie groups on commutative C*-algebras.

math.OA↗
arXiv2022

Compact Group Actions with the Tracial Rokhlin Property

We define a "tracial" analog of the Rokhlin property for actions of second countable compact groups on infinite dimensional simple separable unital C*-algebras. We prove that fixed point algebras under such actions (and, in the appropriate cases, crossed products by such actions) preserve simplicity, Property (SP), tracial rank zero, tracial rank at most one, the Popa property, tracial Jiang-Su stability, Jiang-Su stability when the algebra is nuclear, infiniteness, and pure infiniteness. We also show that the radius of comparison of the fixed point algebra is no larger than that of the original algebra. Our version of the tracial Rokhlin property is an exact generalization of the tracial Rokhlin property for actions of finite groups on classifiable C*-algebras (in the sense of the Elliott program), but for actions of finite groups on more general C*-algebras it may be stronger. We discuss several alternative versions of the tracial Rokhlin property. We give examples of actions of a totally disconnected infinite compact group on a UHF algebra, and of the circle group on a simple unital AT algebra and on the Cuntz algebra ${\mathcal{O}}_{\infty}$, which have our version of the tracial Rokhlin property, but do not have the Rokhlin property, or even finite Rokhlin dimension with commuting towers.

math.OA↗
arXiv2022

Group actions on simple tracially $\mathcal{Z}$-absorbing C*-algebras

We show that if $A$ is a simple (not necessarily unital) tracially $\mathcal{Z}$-absorbing C*-algebra and $α\colon G \to \mathrm{Aut} (A)$ is an action of a finite group $G$ on $A$ with the weak tracial Rokhlin property, then the crossed product $C^*(G, A,α)$ and the fixed point algebra $A^α$ are simple and tracially $\mathcal{Z}$-absorbing, and they are $\mathcal{Z}$-stable if, in addition, $A$ is separable and nuclear. The same conclusion holds for all intermediate C*-algebras of the inclusions $A^α\subseteq A$ and $A \subseteq C^*(G, A,α)$. We prove that if $A$ is a simple tracially $\mathcal{Z}$-absorbing C*-algebra, then, under a finiteness condition, the permutation action of the symmetric group $S_m$ on the minimal $m$-fold tensor product of $A$ has the weak tracial Rokhlin property. We define the weak tracial Rokhlin property for automorphisms of simple C*-algebras and we show that -- under a mild assumption -- (tracial) $\mathcal{Z}$-absorption is preserved under crossed products by such automorphisms.

math.OA↗
arXiv2022

Simple tracially $\mathcal{Z}$-absorbing C*-algebras

We define a notion of tracial $\mathcal{Z}$-absorption for simple not necessarily unital C*-algebras, study it systematically, and prove its permanence properties. This extends the notion defined by Hirshberg and Orovitz for unital C*-algebras. The Razak-Jacelon algebra, simple C*-algebras with tracial rank zero, and simple purely infinite C*-algebras are tracially $\mathcal{Z}$-absorbing. We obtain the first purely infinite examples of tracially $\mathcal{Z}$-absorbing C*-algebras which are not $\mathcal{Z}$-absorbing. We use techniques from reduced free products of von~Neumann algebras to construct these examples. A stably finite example was given by Z. Niu and Q. Wang in 2021. We study the Cuntz semigroup of a simple tracially $\mathcal{Z}$-absorbing C*-algebra and prove that it is almost unperforated and the algebra is weakly almost divisible.

math.OA↗
arXiv2021

The SHAI property for the operators on L^p

A Banach space X has the SHAI (surjective homomorphisms are injective) property provided that for every Banach space Y, every continuous surjective algebra homomorphism from the bounded linear operators on X onto the bounded linear operators on Y is injective. The main result gives a sufficient condition for X to have the SHAI property. The condition is satisfied for L^p (0, 1) for 1 < p < \infty, spaces with symmetric bases that have finite cotype, and the Schatten p-spaces for 1 < p < \infty.

math.FA↗
arXiv2020

Radius of comparison and mean cohomological independence dimension

We introduce a notion of mean cohomological independence dimension for actions of discrete amenable groups on compact metrizable spaces, as a variant of mean dimension, and use it to obtain lower bounds for the radius of comparison of the associated crossed product C*-algebras. Our general theory gives the following for the minimal subshifts constructed by Dou in 2017. Let G be a countable amenable group, let Z be a polyhedron, and let T be Dou's subshift of Z^G (which also depends on a density parameter). Then the radius of comparison of the crossed product is greater than r (1/2) mdim (T) - 2, in which r depends on the density parameter and is close to 1 when the density parameter is close to 1. If Z is even dimensional and has nonvanishing rational cohomology in degree dim (Z), then the radius of comparison of the crossed product is greater than (1/2) mdim (T) - 1, regardless of what the density parameter is.

math.OA↗
arXiv2020

The structure of crossed products by automorphisms of $C (X, D)$

We construct centrally large subalgebras in crossed products of $C (X, D)$ by automorphisms in which $D$ is simple, $X$ is compact metrizable, the automorphism induces a minimal homeomorphism of $X$, and a mild technical assumption holds. We use this construction to prove structural properties of the crossed product, such as (tracial) $Z$-stability, stable rank one, real rank zero, and pure infiniteness, in a number of examples. Our examples are not accessible via methods based on finite Rokhlin dimension, either because $D$ is not $Z$-stable or because $X$ is infinite dimensional.

math.OA↗
arXiv2019

Simplicity of reduced group Banach algebras

Let G be a discrete group. Suppose that the reduced group C*-algebra of G is simple. We use results of Kalantar-Kennedy and Haagerup, and Banach space interpolation, to prove that, for p in (1,infinity), the reduced group L^p operator algebra F^p_r(G) and its *-analog B^{p,*}_r(G) are simple. If G is countable, we prove that the Banach algebras generated by the left regular representations on reflexive Orlicz sequence spaces and certain Lorentz sequence spaces are also simple. We prove analogous results with simplicity replaced by the unique trace property. For use in the Orlicz sequence space case, we prove that if p is in (1,infinity), then any reflexive Orlicz sequence space is isomorphic (not necessarily isometrically) to a space gotten by interpolation between l^p and some other Orlicz sequence space.

math.FA↗
arXiv2019

The Cuntz semigroup and the radius of comparison of the crossed product by a finite group

Let G be a finite group, let A be an infinite-dimensional stably finite simple unital C*-algebra, and let α\colon G \to Aut (A) be an action of G on A which has the weak tracial Rokhlin property. Let A^α be the fixed point algebra. Then the radius of comparison satisfies rc (A^α) \leq rc (A) and rc ( C* (G, A, α) ) \leq ( 1 / card (G) ) rc (A). The inclusion of A^α in A induces an isomorphism from the purely positive part of the Cuntz semigroup Cu (A^α) to the fixed points of the purely positive part of Cu (A), and the purely positive part of Cu ( C* (G, A, α) ) is isomorphic to this semigroup. We construct an example in which G is the two element group, A is a simple unital AH algebra, αhas the Rokhlin property, rc (A) > 0, rc (A^α) = rc (A), and rc (C* (G, A, α)) = (1/2) rc (A).

math.OA↗