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1. CMB Online first

Zhang, Jiao; Wang, Qing-Wen
 An Explicit Formula for the Generalized Cyclic Shuffle Map We provide an explicit formula for the generalized cyclic shuffle map for cylindrical modules. Using this formula we give a combinatorial proof of the generalized cyclic Eilenberg-Zilber theorem. Keywords:generalized Cyclic shuffle map, Cylindrical module, Eilenberg-Zilber theorem, Cyclic homologyCategories:19D55, 05E45

2. CMB 2013 (vol 57 pp. 210)

Zhang, Jiao; Wang, Qing-Wen
 An Explicit Formula for the Generalized Cyclic Shuffle Map We provide an explicit formula for the generalized cyclic shuffle map for cylindrical modules. Using this formula we give a combinatorial proof of the generalized cyclic Eilenberg-Zilber theorem. Keywords:generalized Cyclic shuffle map, Cylindrical module, Eilenberg-Zilber theorem, Cyclic homologyCategories:19D55, 05E45

3. CMB 2012 (vol 56 pp. 551)

Handelman, David
 Real Dimension Groups Dimension groups (not countable) that are also real ordered vector spaces can be obtained as direct limits (over directed sets) of simplicial real vector spaces (finite dimensional vector spaces with the coordinatewise ordering), but the directed set is not as interesting as one would like, i.e., it is not true that a countable-dimensional real vector space that has interpolation can be represented as such a direct limit over the a countable directed set. It turns out this is the case when the group is additionally simple, and it is shown that the latter have an ordered tensor product decomposition. In the Appendix, we provide a huge class of polynomial rings that, with a pointwise ordering, are shown to satisfy interpolation, extending a result outlined by Fuchs. Keywords:dimension group, simplicial vector space, direct limit, Riesz interpolationCategories:46A40, 06F20, 13J25, 19K14

4. CMB 2011 (vol 55 pp. 762)

Li, Hanfeng
 Smooth Approximation of Lipschitz Projections We show that any Lipschitz projection-valued function $p$ on a connected closed Riemannian manifold can be approximated uniformly by smooth projection-valued functions $q$ with Lipschitz constant close to that of $p$. This answers a question of Rieffel. Keywords:approximation, Lipschitz constant, projectionCategory:19K14

5. CMB 2010 (vol 54 pp. 82)

Emerson, Heath
 Lefschetz Numbers for $C^*$-Algebras Using Poincar\'e duality, we obtain a formula of Lefschetz type that computes the Lefschetz number of an endomorphism of a separable nuclear $C^*$-algebra satisfying Poincar\'e duality and the Kunneth theorem. (The Lefschetz number of an endomorphism is the graded trace of the induced map on $\textrm{K}$-theory tensored with $\mathbb{C}$, as in the classical case.) We then examine endomorphisms of Cuntz--Krieger algebras $O_A$. An endomorphism has an invariant, which is a permutation of an infinite set, and the contracting and expanding behavior of this permutation describes the Lefschetz number of the endomorphism. Using this description, we derive a closed polynomial formula for the Lefschetz number depending on the matrix $A$ and the presentation of the endomorphism. Categories:19K35, 46L80

6. CMB 2007 (vol 50 pp. 227)

Kucerovsky, D.; Ng, P. W.
 AF-Skeletons and Real Rank Zero Algebras with the Corona Factorization Property Let $A$ be a stable, separable, real rank zero $C^{*}$-algebra, and suppose that $A$ has an AF-skeleton with only finitely many extreme traces. Then the corona algebra ${\mathcal M}(A)/A$ is purely infinite in the sense of Kirchberg and R\o rdam, which implies that $A$ has the corona factorization property. Categories:46L80, 46L85, 19K35

7. CMB 2007 (vol 50 pp. 268)

Manuilov, V.; Thomsen, K.
 On the Lack of Inverses to $C^*$-Extensions Related to Property T Groups Using ideas of S. Wassermann on non-exact $C^*$-algebras and property T groups, we show that one of his examples of non-invertible $C^*$-extensions is not semi-invertible. To prove this, we show that a certain element vanishes in the asymptotic tensor product. We also show that a modification of the example gives a $C^*$-extension which is not even invertible up to homotopy. Keywords:$C^*$-algebra extension, property T group, asymptotic tensor $C^*$-norm, homotopyCategories:19K33, 46L06, 46L80, 20F99

8. CMB 2005 (vol 48 pp. 221)

Kerr, Matt
 An Elementary Proof of Suslin Reciprocity We state and prove an important special case of Suslin reciprocity that has found significant use in the study of algebraic cycles. An introductory account is provided of the regulator and norm maps on Milnor $K_2$-groups (for function fields) employed in the proof. Categories:19D45, 19E15

9. CMB 2005 (vol 48 pp. 237)

Kimura, Kenichiro
 Indecomposable Higher Chow Cycles Let $X$ be a projective smooth variety over a field $k$. In the first part we show that an indecomposable element in $CH^2(X,1)$ can be lifted to an indecomposable element in $CH^3(X_K,2)$ where $K$ is the function field of 1 variable over $k$. We also show that if $X$ is the self-product of an elliptic curve over $\Q$ then the $\Q$-vector space of indecomposable cycles $CH^3_{ind}(X_\C,2)_\Q$ is infinite dimensional. In the second part we give a new definition of the group of indecomposable cycles of $CH^3(X,2)$ and give an example of non-torsion cycle in this group. Categories:14C25, 19D45

10. CMB 2005 (vol 48 pp. 50)

Elliott, George A.; Gong, Guihua; Li, Liangqing
 Injectivity of the Connecting Maps in AH Inductive Limit Systems Let $A$ be the inductive limit of a system $$A_{1}\xrightarrow{\phi_{1,2}}A_{2} \xrightarrow{\phi_{2,3}} A_{3}\longrightarrow \cd$$ with $A_n = \bigoplus_{i=1}^{t_n} P_{n,i}M_{[n,i]}(C(X_{n,i}))P_{n,i}$, where $~X_{n,i}$ is a finite simplicial complex, and $P_{n,i}$ is a projection in $M_{[n,i]}(C(X_{n,i}))$. In this paper, we will prove that $A$ can be written as another inductive limit $$B_1\xrightarrow{\psi_{1,2}} B_2 \xrightarrow{\psi_{2,3}} B_3\longrightarrow \cd$$ with $B_n = \bigoplus_{i=1}^{s_n} Q_{n,i}M_{\{n,i\}}(C(Y_{n,i}))Q_{n,i}$, where $Y_{n,i}$ is a finite simplicial complex, and $Q_{n,i}$ is a projection in $M_{\{n,i\}}(C(Y_{n,i}))$, with the extra condition that all the maps $\psi_{n,n+1}$ are \emph{injective}. (The result is trivial if one allows the spaces $Y_{n,i}$ to be arbitrary compact metrizable spaces.) This result is important for the classification of simple AH algebras (see \cite{G5,G6,EGL}. The special case that the spaces $X_{n,i}$ are graphs is due to the third named author \cite{Li1}. Categories:46L05, 46L35, 19K14

11. CMB 2004 (vol 47 pp. 431)

Osburn, Robert
 A Note on $4$-Rank Densities For certain real quadratic number fields, we prove density results concerning $4$-ranks of tame kernels. We also discuss a relationship between $4$-ranks of tame kernels and %% $4$-class ranks of narrow ideal class groups. Additionally, we give a product formula for a local Hilbert symbol. Categories:11R70, 19F99, 11R11, 11R45

12. CMB 2003 (vol 46 pp. 509)

Benson, David J.; Kumjian, Alex; Phillips, N. Christopher
 Symmetries of Kirchberg Algebras Let $G_0$ and $G_1$ be countable abelian groups. Let $\gamma_i$ be an automorphism of $G_i$ of order two. Then there exists a unital Kirchberg algebra $A$ satisfying the Universal Coefficient Theorem and with $[1_A] = 0$ in $K_0 (A)$, and an automorphism $\alpha \in \Aut(A)$ of order two, such that $K_0 (A) \cong G_0$, such that $K_1 (A) \cong G_1$, and such that $\alpha_* \colon K_i (A) \to K_i (A)$ is $\gamma_i$. As a consequence, we prove that every $\mathbb{Z}_2$-graded countable module over the representation ring $R (\mathbb{Z}_2)$ of $\mathbb{Z}_2$ is isomorphic to the equivariant $K$-theory $K^{\mathbb{Z}_2} (A)$ for some action of $\mathbb{Z}_2$ on a unital Kirchberg algebra~$A$. Along the way, we prove that every not necessarily finitely generated $\mathbb{Z} [\mathbb{Z}_2]$-module which is free as a $\mathbb{Z}$-module has a direct sum decomposition with only three kinds of summands, namely $\mathbb{Z} [\mathbb{Z}_2]$ itself and $\mathbb{Z}$ on which the nontrivial element of $\mathbb{Z}_2$ acts either trivially or by multiplication by $-1$. Categories:20C10, 46L55, 19K99, 19L47, 46L40, 46L80

13. CMB 2003 (vol 46 pp. 457)

Toms, Andrew
 Strongly Perforated $K_{0}$-Groups of Simple $C^{*}$-Algebras In the sequel we construct simple, unital, separable, stable, amenable $C^{*}$-algebras for which the ordered $K_{0}$-group is strongly perforated and group isomorphic to $Z$. The particular order structures to be constructed will be described in detail below, and all known results of this type will be generalised. Categories:46, 19

14. CMB 2002 (vol 45 pp. 180)

Connolly, Francis X.; Prassidis, Stratos
 On the Exponent of the ${\nk}_0$-Groups of Virtually Infinite Cyclic Groups It is known that the $K$-theory of a large class of groups can be computed from the $K$-theory of their virtually infinite cyclic subgroups. On the other hand, Nil-groups appear to be the obstacle in calculations involving the $K$-theory of the latter. The main difficulty in the calculation of Nil-groups is that they are infinitely generated when they do not vanish. We develop methods for computing the exponent of ${\nk}_0$-groups that appear in the calculation of the $K_0$-groups of virtually infinite cyclic groups. Categories:18F25, 19A31

15. CMB 2000 (vol 43 pp. 37)

Bousaidi, M. A.
 Multiplicative Structure of the Ring $K \bigl( S(T^*\R P^{2n+1}) \bigr)$ We calculate the additive and multiplicative structure of the ring $K\bigl(S(T^*\R P^{2n+1})\bigr)$ using the eta invariant. Categories:19L64, 19K56, 55C35

16. CMB 2000 (vol 43 pp. 69)

Kaminker, Jerome; Perera, Vicumpriya
 Type II Spectral Flow and the Eta Invariant The relative eta invariant of Atiyah-Patodi-Singer will be shown to be expressible in terms of the notion of Type~I and Type~II spectral flow. Categories:19K56, 46L80

17. CMB 1999 (vol 42 pp. 274)

Dădărlat, Marius; Eilers, Søren
 The Bockstein Map is Necessary We construct two non-isomorphic nuclear, stably finite, real rank zero $C^\ast$-algebras $E$ and $E'$ for which there is an isomorphism of ordered groups $\Theta\colon \bigoplus_{n \ge 0} K_\bullet(E;\ZZ/n) \to \bigoplus_{n \ge 0} K_\bullet(E';\ZZ/n)$ which is compatible with all the coefficient transformations. The $C^\ast$-algebras $E$ and $E'$ are not isomorphic since there is no $\Theta$ as above which is also compatible with the Bockstein operations. By tensoring with Cuntz's algebra $\OO_\infty$ one obtains a pair of non-isomorphic, real rank zero, purely infinite $C^\ast$-algebras with similar properties. Keywords:$K$-theory, torsion coefficients, natural transformations, Bockstein maps, $C^\ast$-algebras, real rank zero, purely infinite, classificationCategories:46L35, 46L80, 19K14