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1. CJM Online first

Eilers, Søren; Restorff, Gunnar; Ruiz, Efren; Sørensen, Adam P. W.
Geometric classification of graph C*-algebras over finite graphs
We address the classification problem for graph $C^*$-algebras of finite graphs (finitely many edges and vertices), containing the class of Cuntz-Krieger algebras as a prominent special case. Contrasting earlier work, we do not assume that the graphs satisfy the standard condition (K), so that the graph $C^*$-algebras may come with uncountably many ideals. We find that in this generality, stable isomorphism of graph $C^*$-algebras does not coincide with the geometric notion of Cuntz move equivalence. However, adding a modest condition on the graphs, the two notions are proved to be mutually equivalent and equivalent to the $C^*$-algebras having isomorphic $K$-theories. This proves in turn that under this condition, the graph $C^*$-algebras are in fact classifiable by $K$-theory, providing in particular complete classification when the $C^*$-algebras in question are either of real rank zero or type I/postliminal. The key ingredient in obtaining these results is a characterization of Cuntz move equivalence using the adjacency matrices of the graphs. Our results are applied to discuss the classification problem for the quantum lens spaces defined by Hong and Szymański, and to complete the classification of graph $C^*$-algebras associated to all simple graphs with four vertices or less.

Keywords:graph $C^*$-algebra, geometric classification, $K$-theory, flow equivalence
Categories:46L35, 46L80, 46L55, 37B10

2. CJM 2015 (vol 68 pp. 241)

Allermann, Lars; Hampe, Simon; Rau, Johannes
On Rational Equivalence in Tropical Geometry
This article discusses the concept of rational equivalence in tropical geometry (and replaces an older and imperfect version). We give the basic definitions in the context of tropical varieties without boundary points and prove some basic properties. We then compute the ``bounded'' Chow groups of $\mathbb{R}^n$ by showing that they are isomorphic to the group of fan cycles. The main step in the proof is of independent interest: We show that every tropical cycle in $\mathbb{R}^n$ is a sum of (translated) fan cycles. This also proves that the intersection ring of tropical cycles is generated in codimension 1 (by hypersurfaces).

Keywords:tropical geometry, rational equivalence

3. CJM 2014 (vol 67 pp. 369)

Graham, Robert; Pichot, Mikael
A Free Product Formula for the Sofic Dimension
It is proved that if $G=G_1*_{G_3}G_2$ is free product of probability measure preserving $s$-regular ergodic discrete groupoids amalgamated over an amenable subgroupoid $G_3$, then the sofic dimension $s(G)$ satisfies the equality \[ s(G)=\mathfrak{h}(G_1^0)s(G_1)+\mathfrak{h}(G_2^0)s(G_2)-\mathfrak{h}(G_3^0)s(G_3) \] where $\mathfrak{h}$ is the normalized Haar measure on $G$.

Keywords:sofic groups, dynamical systems, orbit equivalence, free entropy

4. CJM 2014 (vol 67 pp. 184)

McReynolds, D. B.
Geometric Spectra and Commensurability
The work of Reid, Chinburg-Hamilton-Long-Reid, Prasad-Rapinchuk, and the author with Reid have demonstrated that geodesics or totally geodesic submanifolds can sometimes be used to determine the commensurability class of an arithmetic manifold. The main results of this article show that generalizations of these results to other arithmetic manifolds will require a wide range of data. Specifically, we prove that certain incommensurable arithmetic manifolds arising from the semisimple Lie groups of the form $(\operatorname{SL}(d,\mathbf{R}))^r \times (\operatorname{SL}(d,\mathbf{C}))^s$ have the same commensurability classes of totally geodesic submanifolds coming from a fixed field. This construction is algebraic and shows the failure of determining, in general, a central simple algebra from subalgebras over a fixed field. This, in turn, can be viewed in terms of forms of $\operatorname{SL}_d$ and the failure of determining the form via certain classes of algebraic subgroups.

Keywords:arithmetic groups, Brauer groups, arithmetic equivalence, locally symmetric manifolds

5. CJM 2013 (vol 67 pp. 132)

Clouâtre, Raphaël
Unitary Equivalence and Similarity to Jordan Models for Weak Contractions of Class $C_0$
We obtain results on the unitary equivalence of weak contractions of class $C_0$ to their Jordan models under an assumption on their commutants. In particular, our work addresses the case of arbitrary finite multiplicity. The main tool is the theory of boundary representations due to Arveson. We also generalize and improve previously known results concerning unitary equivalence and similarity to Jordan models when the minimal function is a Blaschke product.

Keywords:weak contractions, operators of class $C_0$, Jordan model, unitary equivalence
Categories:47A45, 47L55

6. CJM 2003 (vol 55 pp. 42)

Benanti, Francesca; Di Vincenzo, Onofrio M.; Nardozza, Vincenzo
$*$-Subvarieties of the Variety Generated by $\bigl( M_2(\mathbb{K}),t \bigr)$
Let $\mathbb{K}$ be a field of characteristic zero, and $*=t$ the transpose involution for the matrix algebra $M_2 (\mathbb{K})$. Let $\mathfrak{U}$ be a proper subvariety of the variety of algebras with involution generated by $\bigl( M_2 (\mathbb{K}),* \bigr)$. We define two sequences of algebras with involution $\mathcal{R}_p$, $\mathcal{S}_q$, where $p,q \in \mathbb{N}$. Then we show that $T_* (\mathfrak{U})$ and $T_* (\mathcal{R}_p \oplus \mathcal{S}_q)$ are $*$-asymptotically equivalent for suitable $p,q$.

Keywords:algebras with involution, asymptotic equivalence
Categories:16R10, 16W10, 16R50

7. CJM 1997 (vol 49 pp. 1281)

Sottile, Frank
Pieri's formula via explicit rational equivalence
Pieri's formula describes the intersection product of a Schubert cycle by a special Schubert cycle on a Grassmannian. We present a new geometric proof, exhibiting an explicit chain of rational equivalences from a suitable sum of distinct Schubert cycles to the intersection of a Schubert cycle with a special Schubert cycle. The geometry of these rational equivalences indicates a link to a combinatorial proof of Pieri's formula using Schensted insertion.

Keywords:Pieri's formula, rational equivalence, Grassmannian, Schensted insertion
Categories:14M15, 05E10

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