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1. CJM Online first

Bosa, Joan; Petzka, Henning
Comparison Properties of the Cuntz semigroup and applications to C*-algebras
We study comparison properties in the category $\mathrm{Cu}$ aiming to lift results to the C*-algebraic setting. We introduce a new comparison property and relate it to both the CFP and $\omega$-comparison. We show differences of all properties by providing examples, which suggest that the corona factorization for C*-algebras might allow for both finite and infinite projections. In addition, we show that R{\o}rdam's simple, nuclear C*-algebra with a finite and an infinite projection does not have the CFP.

Keywords:classification of C*-algebras, cuntz semigroup
Categories:46L35, 06F05, 46L05, 19K14

2. CJM 2015 (vol 68 pp. 675)

Martínez-de-la-Vega, Veronica; Mouron, Christopher
Monotone Classes of Dendrites
Continua $X$ and $Y$ are monotone equivalent if there exist monotone onto maps $f:X\longrightarrow Y$ and $g:Y\longrightarrow X$. A continuum $X$ is isolated with respect to monotone maps if every continuum that is monotone equivalent to $X$ must also be homeomorphic to $X$. In this paper we show that a dendrite $X$ is isolated with respect to monotone maps if and only if the set of ramification points of $X$ is finite. In this way we fully characterize the classes of dendrites that are monotone isolated.

Keywords:dendrite, monotone, bqo, antichain
Categories:54F50, 54C10, 06A07, 54F15, 54F65, 03E15

3. CJM 2010 (vol 62 pp. 758)

Dolinar, Gregor; Kuzma, Bojan
General Preservers of Quasi-Commutativity
Let ${ M}_n$ be the algebra of all $n \times n$ matrices over $\mathbb{C}$. We say that $A, B \in { M}_n$ quasi-commute if there exists a nonzero $\xi \in \mathbb{C}$ such that $AB = \xi BA$. In the paper we classify bijective not necessarily linear maps $\Phi \colon M_n \to M_n$ which preserve quasi-commutativity in both directions.

Keywords:general preservers, matrix algebra, quasi-commutativity
Categories:15A04, 15A27, 06A99

4. CJM 2003 (vol 55 pp. 3)

Baake, Michael; Baake, Ellen
An Exactly Solved Model for Mutation, Recombination and Selection
It is well known that rather general mutation-recombination models can be solved algorithmically (though not in closed form) by means of Haldane linearization. The price to be paid is that one has to work with a multiple tensor product of the state space one started from. Here, we present a relevant subclass of such models, in continuous time, with independent mutation events at the sites, and crossover events between them. It admits a closed solution of the corresponding differential equation on the basis of the original state space, and also closed expressions for the linkage disequilibria, derived by means of M\"obius inversion. As an extra benefit, the approach can be extended to a model with selection of additive type across sites. We also derive a necessary and sufficient criterion for the mean fitness to be a Lyapunov function and determine the asymptotic behaviour of the solutions.

Keywords:population genetics, recombination, nonlinear $\ODE$s, measure-valued dynamical systems, Möbius inversion
Categories:92D10, 34L30, 37N30, 06A07, 60J25

5. CJM 2002 (vol 54 pp. 757)

Larose, Benoit
Strongly Projective Graphs
We introduce the notion of strongly projective graph, and characterise these graphs in terms of their neighbourhood poset. We describe certain exponential graphs associated to complete graphs and odd cycles. We extend and generalise a result of Greenwell and Lov\'asz \cite{GreLov}: if a connected graph $G$ does not admit a homomorphism to $K$, where $K$ is an odd cycle or a complete graph on at least 3 vertices, then the graph $G \times K^s$ admits, up to automorphisms of $K$, exactly $s$ homomorphisms to $K$.

Categories:05C15, 06A99

6. CJM 2001 (vol 53 pp. 592)

Perera, Francesc
Ideal Structure of Multiplier Algebras of Simple $C^*$-algebras With Real Rank Zero
We give a description of the monoid of Murray-von Neumann equivalence classes of projections for multiplier algebras of a wide class of $\sigma$-unital simple $C^\ast$-algebras $A$ with real rank zero and stable rank one. The lattice of ideals of this monoid, which is known to be crucial for understanding the ideal structure of the multiplier algebra $\mul$, is therefore analyzed. In important cases it is shown that, if $A$ has finite scale then the quotient of $\mul$ modulo any closed ideal $I$ that properly contains $A$ has stable rank one. The intricacy of the ideal structure of $\mul$ is reflected in the fact that $\mul$ can have uncountably many different quotients, each one having uncountably many closed ideals forming a chain with respect to inclusion.

Keywords:$C^\ast$-algebra, multiplier algebra, real rank zero, stable rank, refinement monoid
Categories:46L05, 46L80, 06F05

7. CJM 1999 (vol 51 pp. 792)

Grätzer, G.; Wehrung, F.
Tensor Products and Transferability of Semilattices
In general, the tensor product, $A \otimes B$, of the lattices $A$ and $B$ with zero is not a lattice (it is only a join-semilattice with zero). If $A\otimes B$ is a {\it capped\/} tensor product, then $A\otimes B$ is a lattice (the converse is not known). In this paper, we investigate lattices $A$ with zero enjoying the property that $A\otimes B$ is a capped tensor product, for {\it every\/} lattice $B$ with zero; we shall call such lattices {\it amenable}. The first author introduced in 1966 the concept of a {\it sharply transferable lattice}. In 1972, H.~Gaskill defined, similarly, sharply transferable semilattices, and characterized them by a very effective condition (T). We prove that {\it a finite lattice $A$ is\/} amenable {\it if{}f it is\/} sharply transferable {\it as a join-semilattice}. For a general lattice $A$ with zero, we obtain the result: {\it $A$ is amenable if{}f $A$ is locally finite and every finite sublattice of $A$ is transferable as a join-semilattice}. This yields, for example, that a finite lattice $A$ is amenable if{}f $A\otimes\FL(3)$ is a lattice if{}f $A$ satisfies (T), with respect to join. In particular, $M_3\otimes\FL(3)$ is not a lattice. This solves a problem raised by R.~W.~Quackenbush in 1985 whether the tensor product of lattices with zero is always a lattice.

Keywords:tensor product, semilattice, lattice, transferability, minimal pair, capped
Categories:06B05, 06B15

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