arXiv subjects

José R. Carrión

Publications and source records attributed to José R. Carrión.

9 recordsLinked to original sources

arXiv2026

Classifying $^$-homomorphisms I: Unital simple nuclear $C^$-algebras

We classify the unital embeddings of a unital separable nuclear $C^*$-algebra satisfying the universal coefficient theorem into a unital simple separable nuclear $C^*$-algebra that tensorially absorbs the Jiang--Su algebra. This gives a new and essentially self-contained proof of the stably finite case of the unital classification theorem: unital simple separable nuclear $C^*$-algebras that absorb the Jiang--Su algebra tensorially and satisfy the universal coefficient theorem are classified by Elliott's invariant of $K$-theory and traces.

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arXiv2026

Tracially Complete C*-Algebras

We introduce a new class of operator algebras -- tracially complete C*-algebras -- as a vehicle for transferring ideas and results between C*-algebras and their tracial von Neumann algebra completions. We obtain structure and classification results for amenable tracially complete C*-algebras satisfying an appropriate version of Murray and von Neumann's property gamma for II_1 factors. In a precise sense, these results fit between Connes' celebrated theorems for injective II_1 factors and the unital classification theorem for separable simple nuclear C*-algebras. The theory also underpins arguments for the known parts of the Toms-Winter conjecture.

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arXiv2026

A Milnor exact sequence for $E$-theory

For separable $C^*$-algebras $A$ and $B$, the $E$-theory group $E(A,B)$ carries a natural, generally non-Hausdorff, second-countable group topology, Hausdorff exactly when the closure of zero $\overline{\{0\}} \subseteq E(A,B)$ is trivial. The nonzero elements of $\overline{\{0\}}$ are the phantom classes, invisible to every continuous Hausdorff-valued invariant. We identify $\overline{\{0\}}$ as a derived inverse limit over any shape system $(C_n)$ of $SA \otimes \mathcal{K}$, giving a natural Milnor sequence $0 \to {\varprojlim}^1 [C_n, S^2B \otimes \mathcal{K}] \to E(A,B) \to \varprojlim [C_n, SB \otimes \mathcal{K}] \to 0$ for all separable $A$ and $B$, with no UCT or nuclearity hypothesis. For nuclear $A$ it agrees with the Willett-Yu controlled-$KK$ Milnor sequence, and when $A$ satisfies the UCT, nuclear or not, its ${\varprojlim}^1$ term is the pure-extension group $\mathrm{Pext}^1_{\mathbb{Z}}(K_{*+1}(A), K_*(B))$, as in Schochet's fine-structure description of the Kasparov groups.

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arXiv2024

A topology on E-theory

For separable $C^*$-algebras $A$ and $B$, we define a topology on the set $[[A, B]]$ consisting of homotopy classes of asymptotic morphisms from $A$ to $B$. This gives an enrichment of the Connes--Higson asymptotic category over topological spaces. We show that the Hausdorffization of this category is equivalent to the shape category of Dadarlat. As an application, we obtain a topology on the $E$-theory group $E(A, B)$ with properties analogous to those of the topology on $KK(A, B)$. The Hausdorffized $E$-theory group $EL(A, B) = E(A, B) / \overline{\{0\}}$ is also introduced and studied. We obtain a continuity result for the functor $EL(\,\cdot\,,B)$, which implies a new continuity result for the functor $KL(\,\cdot\,,B)$.

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arXiv2024

An asymptotic homotopy lifting property

A $C^*$-algebra $A$ is said to have the homotopy lifting property if for all $C^*$-algebras $B$ and $E$, for every surjective $^*$-homomorphism $π\colon E \rightarrow B$ and for every $^*$-homomorphism $ϕ\colon A \rightarrow E$, any path of $^*$-homomorphisms $A \rightarrow B$ starting at $πϕ$ lifts to a path of $^*$-homomorphisms $A \rightarrow E$ starting at $ϕ$. Blackadar has shown that this property holds for all semiprojective $C^*$-algebras. We show that a version of the homotopy lifting property for asymptotic morphisms holds for separable $C^*$-algebras that are sequential inductive limits of semiprojective $C^*$-algebras. It also holds for any separable $C^*$-algebra if the quotient map $π$ satisfies an approximate decomposition property in the spirit of (but weaker than) the notion of quasidiagonality for extensions.

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arXiv2019

Decomposing nuclear maps

We show that the strengthened version of the completely positive approximation property of Brown, Carrión, and White---where the downward maps are asymptotically order zero and the upward maps are convex combinations of order zero maps---is enjoyed by every nuclear order zero map.

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arXiv2017

Decomposable approximations revisited

Nuclear C*-algebras enjoy a number of approximation properties, most famously the completely positive approximation property. This was recently sharpened to arrange for the incoming maps to be sums of order-zero maps. We show that, in addition, the outgoing maps can be chosen to be asymptotically order-zero. Further these maps can be chosen to be asymptotically multiplicative if and only if the C*-algebra and all its traces are quasidiagonal.

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arXiv2015

Almost flat K-theory of classifying spaces

We give a rigorous account and prove continuity properties for the correspondence between almost flat bundles on a triangularizable compact connected space and the quasi-representations of its fundamental group. For a discrete countable group $Γ$ with finite classifying space $BΓ$, we study a correspondence between between almost flat K-theory classes on $BΓ$ and group homomorphism $K_0(C^*(Γ))\to \mathbb{Z}$ that are implemented by pairs of discrete asymptotic homomorphisms from $C^*(Γ)$ to matrix algebras.

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arXiv2013

Quasi-representations of surface groups

By a quasi-representation of a group $G$ we mean an approximately multiplicative map of $G$ to the unitary group of a unital $C^*$-algebra. A quasi-representation induces a partially defined map at the level $K$-theory. In the early 90s Exel and Loring associated two invariants to almost-commuting pairs of unitary matrices $u$ and $v$: one a $K$-theoretic invariant, which may be regarded as the image of the Bott element in $K_0(C(\mathbb{T}^2))$ under a map induced by quasi-representation of $\mathbb{Z}^2$ in U(n); the other is the winding number in $\mathbb{C}\setminus \{0\}$ of the closed path $t\mapsto \det(tvu + (1-t)uv)$. The so-called Exel-Loring formula states that these two invariants coincide if $\|uv - vu\|$ is sufficiently small. A generalization of the Exel-Loring formula for quasi-representations of a surface group taking values in U(n) was given by the second-named author. Here we further extend this formula for quasi-representations of a surface group taking values in the unitary group of a tracial unital $C^*$-algebra.

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