598 research outputs found

    Leavitt RR-algebras over countable graphs embed into L2,RL_{2,R}

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    For a commutative ring RR with unit we show that the Leavitt path algebra LR(E)L_R(E) of a graph EE embeds into L2,RL_{2,R} precisely when EE is countable. Before proving this result we prove a generalised Cuntz-Krieger Uniqueness Theorem for Leavitt path algebras over RR.Comment: 17 pages. At the request of a referee the previous version of this paper has been split into two papers. This version is the first of these papers. The second will also be uploaded to the arXi

    L2,Z⊗L2,ZL_{2,\mathbb{Z}} \otimes L_{2,\mathbb{Z}} does not embed in L2,ZL_{2,\mathbb{Z}}

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    For a commutative ring RR with unit we investigate the embedding of tensor product algebras into the Leavitt algebra L2,RL_{2,R}. We show that the tensor product L2,Z⊗L2,ZL_{2,\mathbb{Z}}\otimes L_{2,\mathbb{Z}} does not embed in L2,ZL_{2,\mathbb{Z}} (as a unital ∗*-algebra). We also prove a partial non-embedding result for the more general L2,R⊗L2,RL_{2,R} \otimes L_{2,R}. Our techniques rely on realising Thompson's group VV as a subgroup of the unitary group of L2,RL_{2,R}.Comment: 16 pages. At the request of a referee the paper arXiv:1503.08705v2 was split into two papers. This is the second of those paper

    The Cuntz splice does not preserve ∗*-isomorphism of Leavitt path algebras over Z\mathbb{Z}

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    We show that the Leavitt path algebras L2,ZL_{2,\mathbb{Z}} and L2−,ZL_{2-,\mathbb{Z}} are not isomorphic as ∗*-algebras. There are two key ingredients in the proof. One is a partial algebraic translation of Matsumoto and Matui's result on diagonal preserving isomorphisms of Cuntz--Krieger algebras. The other is a complete description of the projections in LZ(E)L_{\mathbb{Z}}(E) for EE a finite graph. This description is based on a generalization, due to Chris Smith, of the description of the unitaries in L2,ZL_{2,\mathbb{Z}} given by Brownlowe and the second named author. The techniques generalize to a slightly larger class of rings than just Z\mathbb{Z}.Comment: 17 pages. Since version 2 we extended the arguments from Z to more general ring

    Almost Commuting Orthogonal Matrices

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    We show that almost commuting real orthogonal matrices are uniformly close to exactly commuting real orthogonal matrices. We prove the same for symplectic unitary matrices. This is in contrast to the general complex case, where not all pairs of almost commuting unitaries are close to commuting pairs. Our techniques also yield results about almost normal matrices over the reals and the quaternions.Comment: 13 pages, 3 figure

    Invariance of the Cuntz splice

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    We show that the Cuntz splice induces stably isomorphic graph C∗C^*-algebras.Comment: Our arguments to prove invariance of the Cuntz splice for unital graph C*-algebras in arXiv:1505.06773 applied with only minor changes in the general case. Since most of the results of that preprint have since been superseded by other forthcoming work, we do not intend to publish it, whereas this work is intended for publication. arXiv admin note: substantial text overlap with arXiv:1505.0677

    Amplified graph C*-algebras

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    We provide a complete invariant for graph C*-algebras which are amplified in the sense that whenever there is an edge between two vertices, there are infinitely many. The invariant used is the standard primitive ideal space adorned with a map into {-1,0,1,2,...}, and we prove that the classification result is strong in the sense that isomorphisms at the level of the invariant always lift. We extend the classification result to cover more graphs, and give a range result for the invariant (in the vein of Effros-Handelman-Shen) which is further used to prove that extensions of graph C*-algebras associated to amplified graphs are again graph C*-algebras of amplified graphs.Comment: 15 pages, 1 figur
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