Lajos Molnar
The aim of this paper is to show that between standard operator algebras every bijective map with a certain multiplicativity property related to Jordan triple isomorphisms of associative rings is automatically additive.
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We give a complete description of ring isomorphisms between algebras of measurable operators affiliated with von Neumann algebras of type II$_1.$
We prove that ample groupoids with sigma-compact unit spaces are equivalent if and only if they are stably isomorphic in an appropriate sense, and relate this to Matui's notion of Kakutani equivalence. We use this result to show that diagonal-preserving stable isomorphisms of graph C*-algebras or Leavitt path algebras give rise to isomorphisms of the groupoids of the associated stabilised graphs. We deduce that the Leavitt path algebras $L_Z(E_2)$ and $L_Z(E_{2-})$ are not stably *-isomorphic.
In this paper we verify that the graph forms a complete invariant for Banach algebra isomorphisms of tensor algebras of graphs. For tensor algebras associated with countable directed graphs having no sinks the graph forms an invariant for algebraic isomorphisms as well. The graph also forms a complete invariant for w*-bicontinuous isomorphisms of free semigroupoid algebras. For free semigroupoid algebras associated with countable directed locally finite graphs with no sinks, the graph also forms an invariant for algebraic isomorphisms.
Ring isomorphisms of type II$_\infty$ locally measurable operator algebras
本文证明了II$_\infty$型von Neumann代数的局部可测算子代数间的环同构相似于实$^*$-同构，从而完整描述了一般无有限型I直和项的von Neumann代数对的环同构与格同构。
It is shown that the coloured isomorphism class of a unital, simple, $\mathcal{Z}$-stable, separable amenable C$^*$-algebra satisfying the Universal Coefficient Theorem (UCT) is determined by its tracial simplex.
We construct Hopf algebra isomorphisms of discrete multiplier Hopf C*-algebras, and Hopf AF C*-algebras (generalized quantum UHF algebras), from K-theoretical data. Some of the intermediate results are of independent interest, such as a result that Jordan maps of Hopf algebras intertwine antipodes, and the applications to automorphisms of Hopf algebras.
We give a complete description of order isomorphisms between operator intervals in general von Neumann algebras. For the description, we use Jordan $^*$-isomorphisms and locally measurable operators. Our results generalize several works by L. Molnár and P. Šemrl on type I factors.
We geometrically describe the relation induced on a set of graphs by isomorphism of their associated graph C*-algebras as the smallest equivalence relation generated by five types of moves. The graphs studied have finitely many vertices and finitely or countably infinitely many edges, corresponding to unital and separable C*-algebras.
We resolve the isomorphism problem for tensor algebras of unital multivariable dynamical systems. Specifically we show that unitary equivalence after a conjugation for multivariable dynamical systems is a complete invariant for complete isometric isomorphisms between their tensor algebras. In particular, this settles a conjecture of Davidson and Kakariadis relating to work of Arveson from the sixties, and extends related work of Kakariadis and Katsoulis.
We prove that the isomorphism relation for separable C$^*$-algebras, and also the relations of complete and $n$-isometry for operator spaces and systems, are Borel reducible to the orbit equivalence relation of a Polish group action on a standard Borel space.
We characterize when there exists a diagonal preserving $*$-isomorphism between two graph $C^*$-algebras in terms of the dynamics of the boundary path spaces. In particular, we refine the notion of "orbit equivalence" between the boundary path spaces of the directed graphs $E$ and $F$ and show that this is a necessary and sufficient condition for the existence of a diagonal preserving $*$-isomorphism between the graph $C^*$-algebras $C^*(E)$ and $C^*(F)$.
We study graph $C^*$-algebras equipped with generalised gauge actions, and characterise in terms of groupoids and groupoid cocycles when two graph $C^*$-algebras are isomorphic by a diagonal-preserving isomorphism that intertwines the generalised gauge actions. We apply this characterisation to show that two Cuntz-Krieger algebras are isomorphic by a diagonal-preserving isomorphism that intertwines the gauge actions if and only if the corresponding one-sided subshifts are eventually conjugate, and that the stabilisation of two Cuntz-Krieger algebras are isomorphic by a diagonal-preserving isom…
Rigidity for Isomorphisms between Operator Algebras with Commutative Diagonals
本文证明了多重自由CSL的CSL代数与交换C$^*$-代数的半交叉积这两族算子代数在等距同构下表现出强刚性，即同构类在族内保持不变，且稳定同构等价于等距同构。
We show that if two tensor algebras of topological graphs are algebraically isomorphic, then the graphs are locally conjugate. Conversely, if the base space is at most one dimensional and the edge space is compact, then locally conjugate topological graphs yield completely isometrically isomorphic tensor algebras.
For a Banach D-bimodule M over an abelian unital C*-algebra D, we define E(M) as the collection of norm-one eigenvectors for the dual action of D on the Banach space dual of M. Equip E(M) with the weak-* topology. We develop general properties of E(M). It is properly viewed as a coordinate system for M when M is a subset of C, where C is a unital C*-algebra containing D as a regular MASA with the extension property; moreover, E(C) coincides with Kumjian's twist in the context of C*-diagonals. We identify the C*-envelope of a subalgebra A of a C*-diagonal which contains D. For triangular subalg…
We combine the notion of norming algebra introduced by Pop, Sinclair and Smith with a result of Pisier to show that if A_1 and A_2 are operator algebras, then any bounded epimorphism of A_1 onto A_2 is completely bounded provided that A_2 contains a norming C*-subalgebra. We use this result to give some insights into Kadison's Similarity Problem: we show that every faithful bounded homomorphism of a C*-algebra on a Hilbert space has completely bounded inverse, and show that a bounded representation of a C*-algebra is similar to a *-representation precisely when the image operator algebra λ-nor…
Let $M_q(H_{\mathbb{R}})$ be the $q$-Gaussian von Neumann algebra associated with a separable infinite dimensional real Hilbert space $H_{\mathbb{R}}$ where $-1 < q < 1$. We show that $M_q(H_{\mathbb{R}}) \not \simeq M_0(H_{\mathbb{R}})$ for $-1 < q \not = 0 < 1$. The C$^\ast$-algebraic counterpart of this result was obtained recently in [BCKW22]. Using ideas of Ozawa we show that this non-isomorphism result also holds on the level of von Neumann algebras.
An obstruction to isomorphism of tensor algebras of multivariable dynamical systems
本文通过构造二维空间上具有四个映射的两个分段共轭多变量动力系统，证明其张量代数不同构，从而反驳了 Davidson 和 Katsoulis 的猜想。
Non-isomorphism of $q$-Gaussian von Neumann Algebras in Infinite Variables
本文证明无限维实Hilbert空间对应的$q$-高斯冯·诺依曼代数在$q$取$(-1,1)$内不同值时两两非同构。
Morita equivalence and stable isomorphism via unitary operators
本文通过幺正算子研究双算子系统的Δ-等价与稳定同构，证明其等价关系及在特定情况下的幺正等价，并探讨了正常CES表示的等价性。
Isomorphism rigidity of uniform Roe algebras over arbitrary uniformly locally finite coarse spaces
研究任意一致局部有限粗空间上一致 Roe 代数的同构刚性问题，通过证明\(C^*\) - 代数同构能推出空间双射粗等价，解决了该同构刚性问题。
Lifting isomorphisms in $K$-theory through gradings of $C^*$-algebras
本文证明强Z-分次C*-代数均为Cuntz--Pimsner代数，并利用零分次分量上的K-理论同构判定满射分次同态本身的K-理论同构，同时给出自由阿贝尔群情形下可缩小检验范围的子代数。
The inter-relationship between isomorphisms of commutative and isomorphisms of non-commutative $\log$-algebras
本文研究了非交换 $\log$-代数的同构问题，建立了由不同精确正规半有限迹构造的非交换 $\log$-代数重合及同构的充分必要条件，并揭示了其与 von Neumann 代数中心上 $\log$-代数同构的联系。
Generalizing von Neumann's result on type II$_1$ von Neumann algebras, we characterize lattice isomorphisms between projection lattices of arbitrary von Neumann algebras by means of ring isomorphisms between the algebras of locally measurable operators. Moreover, we give a complete description of ring isomorphisms of locally measurable operator algebras when the von Neumann algebras are without type II direct summands.
We prove that the isomorphism problem for separable nuclear C*-algebras is complete in the class of orbit equivalence relations. In fact, already the isomorphism of simple, separable AI C*-algebras is a complete orbit equivalence relation. This means that any isomorphism problem arsing from a continuous action of a separable completely metrizable group can be reduced to the isomorphism of simple, separable AI C*-algebras. As a consequence, we get that the isomorphism problems for separable nuclear C*-algebras and for separable C*-algebras have the same complexity. This answers questions posed …
Order isomorphisms in $C^*$-algebras
本文研究了$C^*$-代数中自共轭部分和$AW^*$-代数中一般算子区间之间的序同构，利用Jordan $^*$-同构和闭算子进行描述，扩展了冯·诺依曼代数的相关结果。
In this paper a full description of order isomorphisms between effect algebras of atomic JBW-algebras is given. We will derive a closed formula for the order isomorphisms on the effect algebra of type I factors by proving that the invertible part of the effect algebra of a type I factor is left invariant. This yields an order isomorphism on the whole cone, for which a characterisation exists. Furthermore, we will show that the obtained formula for the order isomorphism on the invertible part can be extended to the whole effect algebra again. As atomic JBW-algebras are direct sums of type I fac…
We prove that algebraic isomorphisms between limit algebras are automatically continuous, and consider consequences of this result. In particular, we give partial solutions to a conjecture of Power [Limit Algebras, Longman, 1992, Notes to Chapter 8] and to an open problem [op. cit., Problem 7.8]. As a further consequence, we describe epimorphisms between various classes of limit algebras.
For any countable graph $E$, we investigate the relationship between the Leavitt path algebra $L_{\C}(E)$ and the graph C*-algebra $C^*(E)$. For graphs $E$ and $F$, we examine ring homomorphisms, ring *-homomorphisms, algebra homomorphisms, and algebra *-homomorphisms between $L_{\C}(E)$ and $L_{\C}(F)$. We prove that in certain situations isomorphisms between $L_{\C}(E)$ and $L_{\C}(F)$ yield *-isomorphisms between the corresponding C*-algebras $C^*(E)$ and $C^*(F)$. Conversely, we show that *-isomorphisms between $C^*(E)$ and $C^*(F)$ produce isomorphisms between $L_{\C}(E)$ and $L_{\C}(F)$ …
Constructing number field isomorphisms from *-isomorphisms of certain crossed product C*-algebras
本研究证明了与数域乘法群作用于有限阿代尔环相关的交叉积C*-代数的刚性，即由这类代数的*-同构可构造对应数域的同构，并借此用拓扑全群得到Neukirch–Uchida定理的类似结果，给出一类可完全刻画数域的离散群。
In this paper we completely describe the order isomorphisms between cones of atomic JBW-algebras. Moreover, we can write an atomic JBW-algebra as an algebraic direct summand of the so-called engaged and disengaged part. On the cone of the engaged part every order isomorphism is linear and the disengaged part consists only of copies of $\mathbb{R}$. Furthermore, in the setting of general JB-algebras we prove the following. If either algebra does not contain an ideal of codimension one, then every order isomorphism between their cones is linear if and only if it extends to a homeomorphism, betwe…
We continue the study of isomorphisms of tensor algebras associated to a C*-correspondences in the sense of Muhly and Solel. Inspired by by recent work of Davidson, Ramsey and Shalit, we solve isomorphism problems for tensor algebras arising from weighted partial dynamical systems. We provide complete bounded / isometric classification results for tensor algebras arising from weighted partial systems, both in terms of the C*-correspondences associated to them, and in terms of the original dynamics. We use this to show that the isometric isomorphism and algebraic / bounded isomorphism problems …
The graph groupoids of directed graphs are topologically isomorphic if and only if there is a diagonal-preserving ring *-isomorphism between the Leavitt path algebras.
Work of Jean Renault shows that, for topologically principal étale groupoids, a diagonal-preserving isomorphism of reduced $C^*$-algebras yields an isomorphism of groupoids. Several authors have proved analogues of this result for ample groupoid algebras over integral domains under suitable hypotheses. In this paper, we extend the known results by allowing more general coefficient rings and by weakening the hypotheses on the groupoids. Our approach has the additional feature that we only need to impose conditions on one of the two groupoids. Applications are given to Leavitt path algebras.
Let $K$ be any field, let $L_n$ denote the Leavitt algebra of type $(1,n-1)$ having coefficients in $K$, and let ${\rm M}_d(L_n)$ denote the ring of $d \times d$ matrices over $L_n$. In our main result, we show that ${\rm M}_d(L_n) \cong L_n$ if and only if $d$ and $n-1$ are coprime. We use this isomorphism to answer a question posed in \cite{PS} regarding isomorphisms between various C*-algebras. Furthermore, our result demonstrates that data about the $K_0$ structure is sufficient to distinguish up to isomorphism the algebras in an important class of purely infinite simple $K$-algebras.
Hazrat gave a K-theoretic invariant for Leavitt path algebras as graded algebras. Hazrat conjectured that this invariant classifies Leavitt path algebras up to graded isomorphism, and proved the conjecture in some cases. In this paper, we prove that a weak version of the conjecture holds for all finite essential graphs.
A study of Hilbert $C^*$-bimodules over commutative $C^*$-algebras is carried out and used to establish a sufficient condition for two quantum Heisenberg manifolds to be isomorphic.
We study Steinberg algebras constructed from ample Hausdorff groupoids over commutative integral domains with identity. We reconstruct (graded) groupoids from (graded) Steinberg algebras and use this to characterise when there is a diagonal-preserving (graded) isomorphism between two (graded) Steinberg algebras. We apply this characterisation to groupoids of directed graphs in order to study diagonal-preserving (graded) isomorphisms of Leavitt path algebras and graph $C^*$-algebras.
We consider isomorphisms between quotient algebras of $\prod_{n=0}^{\infty} \mathbb{M}_{k(n)}(\mathbb{C})$ associated with Borel ideals on $\mathbb{N}$ and prove that it is relatively consistent with \textbf{ZFC} that all of these isomorphisms are trivial, in the sense that they lift to a *-homomorphism from $\prod_{n=0}^{\infty} \mathbb{M}_{k(n)}(\mathbb{C})$ into itself. This generalizes a result of Farah-Shelah who proved this result for centers of these algebras (in its dual form). We also use a simpler forcing notion and completely remove the large cardinal assumption used by Farah-Shelah…
In this paper we describe the structure of all sequential isomorphisms between the sets of von Neumann algebra effects. It turns out that if the underlying algebras have no commutative direct summands, then every sequential isomorphism between the sets of their effects extends to the direct sum of a *-isomorphism and a *-antiisomorphism between the underlying von Neumann algebras.
Noncommutative domain algebras were introduced by Popescu as the non-selfadjoint operator algebras generated by weighted shifts on the Full Fock space. This paper uses results from several complex variables to classify many noncommutative domain algebras, and it uses results from operator theory to obtain new bounded domains in hermitian spaces with non-compact automorphic group.
Isomorphisms of Birkhoff-James orthogonality on finite-dimensional $C^*$-algebra
该研究对有限维复$C^*$-代数上强保持Birkhoff-James正交性的双射映射分类，发现其接近实线性等距映射且结构可确定，解决了相关映射的结构刻画问题。
It is shown that all the approximately finite dimensional C*-algebras which are not of Type I are isomorphic as Banach spaces. This generalises the matroid case given previously by Arazy. Analogous results are obtained for various families of triangular subalgebras of AF C*-algebras. In addition the classification of various continua of Type I AF C*-algebras is discussed.
We examine when two maximal abelian algebras in the truncated Toeplitz operators are spatially isomorphic. This builds upon recent work of N. Sedlock, who obtained a complete description of the maximal algebras of truncated Toeplitz operators.
We note that the deep results of Grunewald and Segal on algorithmic problems for arithmetic groups imply the decidability of several matrix equivalence problems involving poset-blocked matrices over Z. Consequently, results of Eilers, Restorff, Ruiz and Sørensen imply that isomorphism and stable isomorphism of unital graph C*-algebras (including the Cuntz-Krieger algebras) are decidable. One can also decide flow equivalence for shifts of finite type, and isomorphism of Z-quiver representations (i.e., finite diagrams of homomorphisms of finitely generated abelian groups).
We prove an algebraic version of the Gauge-Invariant Uniqueness Theorem, a result which gives information about the injectivity of certain homomorphisms between ${\mathbb Z}$-graded algebras. As our main application of this theorem, we obtain isomorphisms between the Leavitt path algebras of specified graphs. From these isomorphisms we are able to achieve two ends. First, we show that the $K_0$ groups of various sets of purely infinite simple Leavitt path algebras, together with the position of the identity element in $K_0$, classifies the algebras in these sets up to isomorphism. Second, we s…
Algebraic isomorphisms of quantized homogeneous spaces
该研究证明了带 Poisson-Lie 稳定子群的单连通紧半单李群齐次空间在不同 $q$ 参数下的量子化正则函数代数彼此不作 $*$-代数同构。
Let ${\rm Rep}\,{\cal O}_n$ denote the category of all nondegenerate $^*$ representations of the Cuntz algebra ${\cal O}_n$. For any $2\leq n,m<\infty$, we construct an isomorphism functor $F_{n,m}$ from ${\rm Rep}\,{\cal O}_m$ to ${\rm Rep}\,{\cal O}_n$.
The hypergraph isomorphism game, Hopf algebras and Galois extensions
本文用紧量子群论建立超图量子同构的代数与操作框架，引入同步博弈代数，证明量子同构异于经典同构，并揭示代数与策略概念的一致性。
Let $\mathcal{G}$ be a generalized matrix algebra over a commutative ring $\mathcal{R}$, ${\mathfrak q}\colon \mathcal{G}\times\mathcal{G}\longrightarrow \mathcal{G}$ be an $\mathcal{R}$-bilinear mapping and ${\mathfrak T}_{\mathfrak q}\colon:\mathcal{G}\longrightarrow \mathcal{G}$ be a trace of $\mathfrak{q}$. We describe the form of ${\mathfrak T}_{\mathfrak q}$ satisfying the condition ${\mathfrak T}_{\mathfrak q}(G)G=G{\mathfrak T}_{\mathfrak q}(G)$ for all $G\in \mathcal{G}$. The question of when ${\mathfrak T}_{\mathfrak q}$ has the proper form is considered. Using the aforementioned tra…
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