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Baojie Jiang

Publications and source records attributed to Baojie Jiang.

7 recordsLinked to original sources
arXiv2026

Exotic Unit Groups of Sylvester Rank Completions

Let \(R\) be any unital ring equipped with a Sylvester matrix rank function, and let \(Q\) be the rank completion of the matrix system defined by block-diagonal embeddings along a factor sequence. We prove that both \(\GL(Q)\) and \((Q,+)\) are exotic: every strongly continuous unitary representation is trivial. We also prove that \(\GL(Q)\) is Bourbaki bounded and admits no nonzero escape function. For every \(\varepsilon>0\), let \(B_\varepsilon\) be the open ball centered at the identity. The least positive integer \(N\) such that \(\GL(Q)=B_\varepsilon^N\) is \(\lfloor\varepsilon^{-1}\rfloor+1\). Neither regularity nor irreducibility is required. Our proof of exoticness uses a uniform Kazhdan estimate for additive integer matrix groups together with rank-one matrix perturbations, rather than the character-theoretic and continuous ring methods used previously. The geometric conclusions follow from factorizations using diagonal corner subgroups.

math.GR↗
arXiv2026

Uniqueness of Rank-Metric Completions of Bratteli Systems

Let $R$ be a unital ring equipped with a Sylvester matrix rank function $\rk$. A harmonic function $α$ on a Bratteli diagram $B$ defines a weighted matrix rank on the associated algebraic direct limit $A(B,R)$. We prove that, if $α$ is extreme and the total weight of blocks of any fixed bounded size tends to zero, then the rank completion of $A(B,R)$ is isomorphic to $\mathcal M_{R,\rk}$, the rank completion of the direct system $\Mat_{2^k}(R)$ with connecting maps $x\mapsto\diag(x,x)$ and normalized ranks $2^{-k}\rk$. The isomorphism preserves the unital $R$-algebra structure and the ranks on all rectangular matrices. The coefficient ring need not be regular, and the specified rank need not be induced from a regular ring. We recover factor-sequence uniqueness and construct corners of every prescribed rank in $(0,1]$ that are isomorphic to $\mathcal M_{R,\rk}$ with their normalized ranks. Examples show that the coefficient rank can affect the isomorphism type and that the completion can be non-regular and non-simple. For complex coefficients, the trace determined by $α$ gives a rank completion of the associated AF $C^*$-algebra canonically isomorphic to the affiliated-operator ring of its GNS closure. The rank completion of the algebraic direct limit can be a proper subring of this ring.

math.RA↗
arXiv2024

Quasi-locality for étale groupoids

Let $\mathcal{G}$ be a locally compact étale groupoid and $\mathscr{L}(L^2(\mathcal{G}))$ be the $C^*$-algebra of adjointable operators on the Hilbert $C^*$-module $L^2(\mathcal{G})$. In this paper, we discover a notion called quasi-locality for operators in $\mathscr{L}(L^2(\mathcal{G}))$, generalising the metric space case introduced by Roe. Our main result shows that when $\mathcal{G}$ is additionally $σ$-compact and amenable, an equivariant operator in $\mathscr{L}(L^2(\mathcal{G}))$ belongs to the reduced groupoid $C^*$-algebra $C^*_r(\mathcal{G})$ if and only if it is quasi-local. This provides a practical approach to describe elements in $C^*_r(\mathcal{G})$ using coarse geometry. Our main tool is a description for operators in $\mathscr{L}(L^2(\mathcal{G}))$ via their slices with the same philosophy to the computer tomography. As applications, we recover a result by Špakula and the second-named author in the metric space case, and deduce new characterisations for reduced crossed products and uniform Roe algebras for groupoids.

math.OA↗
arXiv2024

Rigidity for geometric ideals in uniform Roe algebras

In this paper, we investigate the rigidity problems for geometric ideals in uniform Roe algebras associated to discrete metric spaces of bounded geometry. These ideals were introduced by Chen and Wang, and can be fully characterised in terms of ideals in the associated coarse structures. Our main result is that if two geometric ideals in uniform Roe algebras are stably isomorphic, then the coarse spaces associated to these ideals are coarsely equivalent. We also discuss the case of ghostly ideals and pose some open questions.

math.OA↗
arXiv2020

Sylvester rank functions for amenable normal extensions

We introduce a notion of amenable normal extension S of a unital ring R with a finite approximation system F, encompassing the amenable algebras over a field of Gromov and Elek, the twisted crossed product by an amenable group, and the tensor product with a field extension. It is shown that every Sylvester matrix rank function rk of R preserved by S has a canonical extension to a Sylvester matrix rank function rk_F for S. In the case of twisted crossed product by an amenable group, and the tensor product with a field extension, it is also shown that rk_F depends on rk continuously. When an amenable group has a twisted action on a unital C*-algebra preserving a tracial state, we also show that two natural Sylvester matrix rank functions on the algebraic twisted crossed product constructed out of the tracial state coincide.

math.RA↗
arXiv2015

Property $T$ of reduced $C^*$-crossed products by discrete groups

We generalize the main result of Kamalov and show that if $G$ is an amenable discrete group with an action $α$ on a finite nuclear unital $C^*$-algebra $A$ such that the reduced crossed product $A\rtimes_{α,r} G$ has property $T$, then $G$ is finite and $A$ is finite dimensional. As an application, an infinite discrete group $H$ is non-amenable if and only if the uniform Roe algebra $C^*_u(H)$ has property $T$.

math.OA↗