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Huaxin Lin

Publications and source records attributed to Huaxin Lin.

At least 19 recordsLinked to original sources
arXiv2025

Approximately Macroscopically Unique States and Quantum Mechanics

We show that Mumford's Approximately Macroscopically Unique (AMU) states exist for quantum systems consisting of unbounded self-adjoint operators when the commutators are small. In particular, AMU states always exist in position and momentum systems when the Planck constant $|\hbar|$ is sufficiently small. However, we show that these standard quantum mechanical systems are far away from classical mechanical (commutative) systems even when $|\hbar|\to 0.$

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arXiv2025

Almost Representations

Let $H$ be an infinite dimensional separable Hilbert space, $B(H)$ the $C^*$-algebra of all bounded linear operators on $H,$ $U(B(H))$ the unitary group of $B(H)$ and ${\cal K}\subset B(H)$ the ideal of compact operators. Let $G$ be a countable discrete amenable group. We prove the following: For any $\epsilon>0,$ any finite subset ${\cal F}\subset G,$ and $0<\sigma\le 1,$ there exists $\delta>0,$ finite subsets ${\cal G}\subset G$ and ${\cal S}\subset {\bf C}[G]$ satisfying the following property: For any map $\phi: G\to U(B(H))$ such that $$ \|\phi(fg)-\phi(f)\phi(g)\|<\delta\,\,\,for\,\, all\,\, f,g\in {\cal G}\,\,\, and \,\,\, \|\pi\circ \tilde \phi(x)\|\ge \sigma \|x\|\,\,\, for\,\, all\,\, x\in {\cal S}, $$ there is a group homomorphism $h: G\to U(B(H))$ such that $$ \|\phi(f)-h(f)\|<\epsilon\,\,\, for\,\,\, all\,\,\, f\in {\cal F}, $$ where $\tilde \phi$ is the linear extension of $\phi$ on the group ring ${\bf C}[G]$ and $\pi: B(H)\to B(H)/{\cal K}$ is the quotient map. A counterexample is given that the fullness condition above cannot be removed. We actually prove a more general result for separable amenable $C^*$-algebras.

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arXiv2024

Existence of Approximately Macroscopically Unique States

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the C*-algebra of bounded operators on $H.$ Suppose that $T_1,T_2,..., T_n$ are self-adjoint operators in $B(H).$ We show that, if commutators $[T_i, T_j]$ are sufficiently small in norm, then ``Approximately Macroscopically Unique" states always exist for any values in a synthetic spectrum of the $n$-tuple of self-adjoint operators. This is achieved under the circumstance for which the $n$-tuple may not be approximated by commuting ones. This answers a question proposed by David Mumford for measurements in quantum theory. If commutators are not small in norm but small modulo compact operators, then ``Approximate Macroscopic Uniqueness" states also exist.

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arXiv2024

Almost commuting self-adjoint operators and measurements

We study the problem when an almost commuting $n$-tuple self-adjoint operators in an infinite dimensional separable Hilbert space $H$ is close to an $n$-tuple of commuting self-adjoint operators on $H.$ We give an affirmative answer to the problem when the synthetic-spectrum and the essential synthetic-spectrum are close. Examples are also exhibited that, in general, the answer to the problem when $n\ge 3$ is negative even the associated Fredholm index vanishes. In the case that $n=2,$ we show that a pair of almost commuting self-adjoint operators in an infinite dimensional separable Hilbert space is close to a commuting pair of self-adjoint operators if and only if a corresponding Fredholm index vanishes outside of an essential synthetic-spectrum. This is an attempt to solve a problem proposed by David Mumford related to quantum theory and measurements.

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arXiv2023

Double duals and Hilbert modules

Let $A$ be a $C^*$-algebra, $H$ be a Hilbert $A$-module and $K(H)$ be the closure of the set of finite rank module maps. We show that the $W^*$-algebra of all bounded $A^{**}$-module maps on the smallest self-dual Hilbert $A^{**}$-module containing $H$ is isomorphic to $K(H)^{**}$ as $W^*$-algebras. We also show that the unit ball of $H$ is closed in $H^\sharp,$ the dual of $H,$ in an $A$-weak topology of $H^\sharp$ as well as dense in the unit ball of $H^\sharp$ in a weak*-topology and some versions of Kaplansky density theorem for Hilbert $C^*$-modules.

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arXiv2023

Extensions of C*-algebras

Let $A$ be a separable amenable $C^*$-algebra and $B$ a non-unital and $\sigma$-unital simple $C^*$-algebra with continuous scale ($B$ need not be stable). We classify, up to unitary equivalence, all essential extensions of the form $0 \rightarrow B \rightarrow D \rightarrow A \rightarrow 0$ using KK theory. There are characterizations of when the relation of weak unitary equivalence is the same as the relation of unitary equivalence, and characterizations of when an extension is liftable (a.k.a.~trivial or split). In the case where $B$ is purely infinite, an essential extension $\rho : A \rightarrow M(B)/B$ is liftable if and only if $[\rho]=0$ in $KK(A, M(B)/B)$. When $B$ is stably finite, the extension $\rho$ is often not liftable when $[\rho]=0$ in $KK(A, M(B)/B).$ Finally, when $B$ additionally has tracial rank zero and when $A$ belongs to a sufficiently regular class of unital separable amenable $C^*$-algebras, we have a version of the Voiculescu noncommutative Weyl--von Neumann theorem: Suppose that $\Phi, \Psi: A \rightarrow M(B)$ are unital injective homomorphisms such that $\Phi(A) \cap B = \Psi(A) \cap B = \{ 0 \}$ and $\tau \circ \Phi = \tau \circ \Psi$ for all $\tau \in T(B),$ {the tracial state space of $B.$} Then there exists a sequence $\{ u_n \}$ of unitaries in $M(B)$ such that (i) $u_n \Phi(a) u_n^* - \Psi(a) \in B$ for all $a \in A$ and $n \geq 1$, (ii) $\| u_n \Phi(a) u_n^* - \Psi(a) \| \rightarrow 0$ as $n \rightarrow \infty$ for all $a \in A$.

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arXiv2023

Strict comparison and stable rank one

Let $A$ be a $\sigma$-unital finite simple $C^*$-algebra which has strict comparison property. We show that if the canonical map $\Gamma$ from the Cuntz semigroup to certain lower semi-continuous affine functions is surjective, then $A$ has tracial approximate oscillation zero and stable rank one. Equivalently, if $A$ has an almost unperforated and almost divisible Cuntz semigroup, then $A$ has stable rank one and tracial approximate oscillation zero.

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arXiv2023

Projective Hilbert Modules and Sequential Approximation

We show that, when $A$ is a separable C*-algebra, every countably generated Hilbert $A$-module is projective (with bounded module maps as morphisms). We also study the approximate extensions of bounded module maps. In the case that $A$ is a $\sigma$-unital simple C*-algebra with strict comparison and every strictly positive lower semicontinuous affine function on quasitraces can be realized as the rank of an element in Cuntz semigroup, we show that the Cuntz semigroup is the same as unitarily equivalent class of countably generated Hilbert $A$-modules if and only if $A$ has stable rank one.

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arXiv2022

Tracial approximation and ${\cal Z}$-stability

Let $A$ be a unital separable non-elementary amenable simple stably finite C*-algebra such that its tracial state space has a $\sigma$-compact countable-dimensional extremal boundary. We show that $A$ is ${\cal Z}$-stable if and only if it has strict comparison and stable rank one. We show that this result also holds for non-unital cases (which may not be Morita equivalent to unital ones).

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arXiv2022

Hereditary uniform property $\Gamma$

We study the uniform property $\Gamma$ for separable simple $C^*$-algebras which have quasitraces and may not be exact. We show that a stably finite separable simple $C^*$-algebra $A$ with strict comparison and uniform property $\Gamma$ has tracial approximate oscillation zero and stable rank one. Moreover in this case, its hereditary $C^*$-subalgebras also have a version of uniform property $\Gamma.$ If a separable non-elementary simple amenable $C^*$-algebra $A$ with strict comparison has this hereditary uniform property $\Gamma,$ then $A$ is ${\cal Z}$-stable.

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arXiv2021

Tracial oscillation zero and stable rank one

Let $A$ be a separable (not necessarily unital) simple $C^*$-algebra with strict comparison. We show that if $A$ has tracial approximate oscillation zero then $A$ has stable rank one and the canonical map $\Gamma$ from the Cuntz semigroup of $A$ to the corresponding affine function space is surjective. The converse also holds. As a by-product, we find that a separable simple $C^*$-algebra which has almost stable rank one must have stable rank one, provided it has strict comparison and the canonical map $\Gamma$ is surjective.

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arXiv2021

Tracial oscillation zero and Z-stability

Let $A$ be a (not necessarily unital) separable non-elementary simple amenable C*-algebra whose tracial basis may not have finite covering dimension and may not be compact but satisfies certain condition (C). We show that $A$ is ${\cal Z}$-stable if and only if $A$ has strict comparison for positive elements. Extremal boundaries of simplexes which satisfy condition (C) may contain countable disjoint unions of $n$-dimensional cubes ($n\in \N$) as a subset.

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arXiv2021

Tracial approximate divisibility and stable rank one

We show that every separable simple tracially approximately divisible $C^*$-algebra has strict comparison, is either purely infinite, or has stable rank one. As a consequence, we show that every (non-unital) finite simple ${\cal Z}$-stable $C^*$-algebra has stable rank one.

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arXiv2021

Non-amenable simple C*-algebras with tracial approximation

We construct two types of unital separable simple $C^*$-alebras $A_z^{C_1}$ and $A_z^{C_2},$ one is exact but not amenable, and the other is non-exact. Both have the same Elliott invariant as the Jiang-Su algebra, namely, $A_z^{C_i}$ has a unique tracial state, $$(K_0(A_z^{C_i}), K_0(A_z^{C_i})_+, [1_{A_z^{C_i}} ])=(\mathbb Z, \mathbb Z_+,1)$$ and $K_{1}(A_z^{C_i})=\{0\}$ ($i=1,2$). We show that $A_z^{C_i}$ ($i=1,2$) is essentially tracially in the class of separable ${\cal Z}$-stable $C^*$-alebras of nuclear dimension 1. $A_z^{C_i}$ has stable rank one, strict comparison for positive elements and no 2-quasitrace other than the unique tracial state. We also produce models of unital separable simple non-exact $C^*$-alebras which are essentially tracially in the class of simple separable nuclear ${\cal Z}$-stable $C^*$-alebras and the models exhaust all possible weakly unperforated Elliott invariants. We also discuss some basic properties of essential tracial approximation.

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arXiv2020

Unitary groups and augmented Cuntz semigroups of separable simple Z-stable C*-algebras

Let $A$ be a separable simple exact ${\cal Z}$-stable $C^*$-algebra. We show that the unitay group of ${\tilde A}$ has the cancellation property. If $A$ has continuous scale, the Cuntz semigroup of $\tilde A$ has the strict comparison property and a weak cancellation property. Let $C$ be a 1-dimensional non-commutative CW complex with $K_1(C)=\{0\}.$ Suppose that $\lambda: {\rm Cu}^\sim(C)\to {\rm Cu}^\sim(A)$ is a morphism in Cuntz semigroups which is strictly positive. Then there exists a sequence of homomorphisms $\phi_n: C\to A$ such that $\lim_{n\to\infty}{\rm Cu}^\sim(\phi_n)=\lambda.$ This result leads to the proof that every separable amenable simple $C^*$-algebra in the UCT class has rationally generalized tracial rank at most one.

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arXiv2020

On classification of non-unital amenable simple C*-algebras, III, the range and the reduction

Following Elliott's earlier work, we show that the Elliott invariant of any finite separable simple $C^*$-algebra with finite nuclear dimension can always be described as a scaled simple ordered group pairing together with a countable abelian group which unifies the unital and nonunital, as well as stably projectionless cases. We also show that, for any given such invariant set, there is a finite separable simple $C^*$-algebra, whose Elliott invariant is the given set, a refinement of the range theorem of Elliott in the stable case. In the stably projectionless case, modified model $C^*$-algebras are constructed in such a way that they are of generalized tracial rank one and have other technical features. We also show that every stably projectionless separable simple amenable $C^*$-algebra in the UCT class has rationally generalized tracial rank one.

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