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Yeong Chyuan Chung

Publications and source records attributed to Yeong Chyuan Chung.

13 recordsLinked to original sources
arXiv2026

Ideal structure of $\ell^p$ uniform Roe algebras

For a uniformly locally finite coarse space $(X,\mathcal E)$ and $p\in\{0\}\cup[1,\infty]$, we classify the nonzero geometric ideals of the $\ell^p$ uniform Roe algebra $B_u^p(X,\mathcal E)$ by the ideals of $\mathcal E$, equivalently by ideals in the associated family of controlled partial coverings. Hence the geometric ideal lattice is canonically independent of $p$. For $p\in[1,\infty]$, we identify $B_u^p(X,\mathcal E)$ isometrically with the reduced $L^p$ operator algebra of the coarse groupoid and show that this identification preserves essential support, carrying geometric and ghostly ideals to dynamical and restrictive ideals. Under property A, controlled $p$-kernels yield a multiplier approximate identity with controlled propagation; consequently every ideal is geometric and every ghost is trivial. Thus the full ideal lattices are canonically independent of $p$ under property A. For $p\in\{0,1,\infty\}$, the complete ideal classification holds without property A; on the groupoid side, the corresponding unconditional conclusion holds for $p\in\{1,\infty\}$.

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arXiv2026

Morita Induction and Preservation of Geometric and Dynamical Ideals

Given Banach algebras with bounded approximate identities that are Morita equivalent in the sense of Paravicini, we give explicit formulas for the induced correspondence of closed ideals, characterize it via localized Morita submodules, determine when corresponding ideals are Morita equivalent via restricted bimodules, and show that corresponding quotients are always Morita equivalent. We prove preservation of geometric cores associated to compatible dense algebraic cores. For equivalent étale groupoids with paracompact unit spaces, Morita induction preserves dynamical ideals and inner supports. For the Morita equivalence between the $\ell^p$ uniform Roe algebra and the $\ell^p$ uniform algebra, we prove that the latter has a bounded approximate identity of projections, identify its geometric ideals explicitly, and identify corresponding complete-lattice block decompositions of the full ideal lattices with matching extremal ideals. Under property A, or for $p=1$, these blocks collapse; for $p=2$, collapse of the minimal block characterizes property A.

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arXiv2026

On some $p$-approximation properties of discrete groups and $\ell^p$ uniform Roe algebras

We study $p$-approximation properties of $\ell^p$ uniform Roe algebras and their connections to coarse geometry and group theory. For bounded geometry metric spaces, property A implies $p$-nuclearity of $B_u^p(X)$ for $1<p<\infty$, while $B_u^1(X)$ is always $1$-nuclear. Property A also yields coefficient characterizations and $p$-complete exactness results for $B_u^p(X,-)$; at $p=1$ these coefficient results hold without property A. We introduce the $p$-invariant translation approximation property and its operator space version. The $p$-AP implies a coefficient fixed-point identity for arbitrary $SQ_p$ coefficients, and for exact groups it is equivalent to the $p$-operator ITAP. At $p=1$, every discrete group satisfies the full coefficient fixed-point identity. We also establish coefficient permanence under subgroups, finite-index supergroups, and group extensions with $p$-AP quotient, including direct and semidirect products. In particular, the class of exact groups with the $p$-AP is closed under group extensions. We also obtain higher-rank lattices as examples failing the $p$-operator ITAP.

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arXiv2026

Morita equivalence for $L^p$-operator algebras associated with étale groupoids

We prove that equivalent locally compact, locally Hausdorff, étale groupoids with paracompact unit spaces have Morita equivalent reduced and full $L^p$-operator algebras for every $p\in[1,\infty]$. The reduced theorem requires pairing-valued approximate identities, reflecting the failure of unconditionality of the reduced norm for $1<p<\infty$, while the full theorem rests on a full-clopen reduction theorem proved by dilating spatial representations. For $1\leq p<\infty$, we also compare the concrete $p$-linking algebra of the reduced Morita equivalence with the reduced $L^p$-operator algebra of the linking groupoid: the canonical comparison is contractive, injective, and has dense range in general, and is an isometric isomorphism for $p=2$. We further study Morita cycles arising from proper étale groupoid correspondences, including the reduced case under suitable extension hypotheses. Applications include an $L^p$ version of Green's symmetric imprimitivity theorem, results for coarse groupoids and inverse semigroups, and invariance of the corresponding $K$-theory.

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arXiv2024

Quasi-countable inverse semigroups as metric spaces, and the uniform Roe algebras of locally finite inverse semigroups

Given any quasi-countable, in particular any countable inverse semigroup $S$, we introduce a way to equip $S$ with a proper and right subinvariant extended metric. This generalizes the notion of proper, right invariant metrics for discrete countable groups. Such a metric is shown to be unique up to bijective coarse equivalence of the semigroup, and hence depends essentially only on $S$. This allows us to unambiguously define the uniform Roe algebra of $S$, which we prove can be realized as a canonical crossed product of $\ell^\infty(S)$ and $S$. We relate these metrics to the analogous metrics on Hausdorff étale groupoids. Using this setting, we study those inverse semigroups with asymptotic dimension $0$. Generalizing results known for groups, we show that these are precisely the locally finite inverse semigroups, and are further characterized by having strongly quasi-diagonal uniform Roe algebras. We show that, unlike in the group case, having a finite uniform Roe algebra is strictly weaker and is characterized by $S$ being locally $\mathcal L$-finite, and equivalently sparse as a metric space.

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arXiv2023

Morita equivalence of two $\ell^p$ Roe-type algebras

Given a metric space with bounded geometry, one may associate with it the $\ell^p$ uniform Roe algebra and the $\ell^p$ uniform algebra, both containing information about the large scale geometry of the metric space. We show that these two Banach algebras are Morita equivalent in the sense of Lafforgue for $1\leq p<\infty$. As a consequence, these two Banach algebras have the same $K$-theory. We then define an $\ell^p$ uniform coarse assembly map taking values in the $K$-theory of the $\ell^p$ uniform Roe algebra and show that it is not always surjective.

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arXiv2021

Property A and coarse embeddability for fuzzy metric spaces

Property A is a geometric property originally introduced for discrete metric spaces to provide a sufficient condition for coarse embeddability into Hilbert space, and it is defined via a Følner condition similar in spirit to the classical notion of amenability for groups. In this paper, we define property A for fuzzy metric spaces in the sense of George and Veeramani, show that it is an invariant in the coarse category of fuzzy metric spaces, and provide characterizations of it for uniformly locally finite fuzzy metric spaces. We also show that uniformly locally finite fuzzy metric spaces with property A are coarsely embeddable into Hilbert space.

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arXiv2020

Coarse Baum-Connes conjecture and rigidity for Roe algebras

In this paper, we connect the rigidity problem and the coarse Baum-Connes conjecture for Roe algebras. In particular, we show that if $X$ and $Y$ are two uniformly locally finite metric spaces such that their Roe algebras are $*$-isomorphic, then $X$ and $Y$ are coarsely equivalent provided either $X$ or $Y$ satisfies the coarse Baum-Connes conjecture with coefficients. It is well-known that coarse embeddability into a Hilbert space implies the coarse Baum-Connes conjecture with coefficients. On the other hand, we provide a new example of a finitely generated group satisfying the coarse Baum-Connes conjecture with coefficients but which does not coarsely embed into a Hilbert space.

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arXiv2019

Dynamical Complexity and $K$-Theory of $L^p$ Operator Crossed Products

We apply quantitative (or controlled) $K$-theory to prove that a certain $L^p$ assembly map is an isomorphism for $p\in[1,\infty)$ when an action of a countable discrete group $Γ$ on a compact Hausdorff space $X$ has finite dynamical complexity. When $p=2$, this is a model for the Baum-Connes assembly map for $Γ$ with coefficients in $C(X)$, and was shown to be an isomorphism by Guentner, Willett, and Yu.

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arXiv2019

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 $p\in [1,\infty)$.

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arXiv2018

Rigidity of $\ell^p$ Roe-type algebras

We investigate the rigidity of the $\ell^p$ analog of Roe-type algebras. In particular, we show that if $p\in[1,\infty)\setminus\{2\}$, then an isometric isomorphism between the $\ell^p$ uniform Roe algebras of two metric spaces with bounded geometry yields a bijective coarse equivalence between the underlying metric spaces, while a stable isometric isomorphism yields a coarse equivalence. We also obtain similar results for other $\ell^p$ Roe-type algebras. In this paper, we do not assume that the metric spaces have Yu's property A or finite decomposition complexity.

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arXiv2017

Quantitative $K$-theory for Banach algebras

Quantitative (or controlled) $K$-theory for $C^*$-algebras was used by Guoliang Yu in his work on the Novikov conjecture, and later developed more formally by Yu together with Hervé Oyono-Oyono. In this paper, we extend their work by developing a framework of quantitative $K$-theory for the class of algebras of bounded linear operators on subquotients (i.e., subspaces of quotients) of $L_p$ spaces. We also prove the existence of a controlled Mayer-Vietoris sequence in this framework.

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