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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 2016 (vol 68 pp. 1023)

Phillips, John; Raeburn, Iain
Centre-valued Index for Toeplitz Operators with Noncommuting Symbols
We formulate and prove a ``winding number'' index theorem for certain ``Toeplitz'' operators in the same spirit as Gohberg-Krein, Lesch and others. The ``number'' is replaced by a self-adjoint operator in a subalgebra $Z\subseteq Z(A)$ of a unital $C^*$-algebra, $A$. We assume a faithful $Z$-valued trace $\tau$ on $A$ left invariant under an action $\alpha:{\mathbf R}\to Aut(A)$ leaving $Z$ pointwise fixed.If $\delta$ is the infinitesimal generator of $\alpha$ and $u$ is invertible in $\operatorname{dom}(\delta)$ then the ``winding operator'' of $u$ is $\frac{1}{2\pi i}\tau(\delta(u)u^{-1})\in Z_{sa}.$ By a careful choice of representations we extend $(A,Z,\tau,\alpha)$ to a von Neumann setting $(\mathfrak{A},\mathfrak{Z},\bar\tau,\bar\alpha)$ where $\mathfrak{A}=A^{\prime\prime}$ and $\mathfrak{Z}=Z^{\prime\prime}.$ Then $A\subset\mathfrak{A}\subset \mathfrak{A}\rtimes{\bf R}$, the von Neumann crossed product, and there is a faithful, dual $\mathfrak{Z}$-trace on $\mathfrak{A}\rtimes{\bf R}$. If $P$ is the projection in $\mathfrak{A}\rtimes{\bf R}$ corresponding to the non-negative spectrum of the generator of $\mathbf R$ inside $\mathfrak{A}\rtimes{\mathbf R}$ and $\tilde\pi:A\to\mathfrak{A}\rtimes{\mathbf R}$ is the embedding then we define for $u\in A^{-1}$, $T_u=P\tilde\pi(u) P$ and show it is Fredholm in an appropriate sense and the $\mathfrak{Z}$-valued index of $T_u$ is the negative of the winding operator. In outline the proof follows the proof of the scalar case done previously by the authors. The main difficulty is making sense of the constructions with the scalars replaced by $\mathfrak{Z}$ in the von Neumann setting. The construction of the dual $\mathfrak{Z}$-trace on $\mathfrak{A}\rtimes{\mathbf R}$ required the nontrivial development of a $\mathfrak{Z}$-Hilbert Algebra theory. We show that certain of these Fredholm operators fiber as a ``section'' of Fredholm operators with scalar-valued index and the centre-valued index fibers as a section of the scalar-valued indices.

Keywords:index ,Toeplitz operator
Categories:46L55, 19K56, 46L80

3. CJM 2015 (vol 68 pp. 150)

Stavrova, Anastasia
Non-stable $K_1$-functors of Multiloop Groups
Let $k$ be a field of characteristic 0. Let $G$ be a reductive group over the ring of Laurent polynomials $R=k[x_1^{\pm 1},...,x_n^{\pm 1}]$. Assume that $G$ contains a maximal $R$-torus, and that every semisimple normal subgroup of $G$ contains a two-dimensional split torus $\mathbf{G}_m^2$. We show that the natural map of non-stable $K_1$-functors, also called Whitehead groups, $K_1^G(R)\to K_1^G\bigl( k((x_1))...((x_n)) \bigr)$ is injective, and an isomorphism if $G$ is semisimple. As an application, we provide a way to compute the difference between the full automorphism group of a Lie torus (in the sense of Yoshii-Neher) and the subgroup generated by exponential automorphisms.

Keywords:loop reductive group, non-stable $K_1$-functor, Whitehead group, Laurent polynomials, Lie torus
Categories:20G35, 19B99, 17B67

4. CJM 2013 (vol 65 pp. 1287)

Reihani, Kamran
$K$-theory of Furstenberg Transformation Group $C^*$-algebras
The paper studies the $K$-theoretic invariants of the crossed product $C^{*}$-algebras associated with an important family of homeomorphisms of the tori $\mathbb{T}^{n}$ called Furstenberg transformations. Using the Pimsner-Voiculescu theorem, we prove that given $n$, the $K$-groups of those crossed products, whose corresponding $n\times n$ integer matrices are unipotent of maximal degree, always have the same rank $a_{n}$. We show using the theory developed here that a claim made in the literature about the torsion subgroups of these $K$-groups is false. Using the representation theory of the simple Lie algebra $\frak{sl}(2,\mathbb{C})$, we show that, remarkably, $a_{n}$ has a combinatorial significance. For example, every $a_{2n+1}$ is just the number of ways that $0$ can be represented as a sum of integers between $-n$ and $n$ (with no repetitions). By adapting an argument of van Lint (in which he answered a question of Erdős), a simple, explicit formula for the asymptotic behavior of the sequence $\{a_{n}\}$ is given. Finally, we describe the order structure of the $K_{0}$-groups of an important class of Furstenberg crossed products, obtaining their complete Elliott invariant using classification results of H. Lin and N. C. Phillips.

Keywords:$K$-theory, transformation group $C^*$-algebra, Furstenberg transformation, Anzai transformation, minimal homeomorphism, positive cone, minimal homeomorphism
Categories:19K14, 19K99, 46L35, 46L80, , 05A15, 05A16, 05A17, 15A36, 17B10, 17B20, 37B05, 54H20

5. CJM 2012 (vol 64 pp. 368)

Meyer, Ralf; Nest, Ryszard
C$^*$-Algebras over Topological Spaces: Filtrated K-Theory
We define the filtrated K-theory of a $\mathrm{C}^*$-algebra over a finite topological space \(X\) and explain how to construct a spectral sequence that computes the bivariant Kasparov theory over \(X\) in terms of filtrated K-theory. For finite spaces with a totally ordered lattice of open subsets, this spectral sequence becomes an exact sequence as in the Universal Coefficient Theorem, with the same consequences for classification. We also exhibit an example where filtrated K-theory is not yet a complete invariant. We describe two $\mathrm{C}^*$-algebras over a space \(X\) with four points that have isomorphic filtrated K-theory without being $\mathrm{KK}(X)$-equivalent. For this space \(X\), we enrich filtrated K-theory by another K-theory functor to a complete invariant up to $\mathrm{KK}(X)$-equivalence that satisfies a Universal Coefficient Theorem.

Keywords:46L35, 46L80, 46M18, 46M20

6. CJM 2011 (vol 63 pp. 381)

Ji, Kui ; Jiang, Chunlan
A Complete Classification of AI Algebras with the Ideal Property
Let $A$ be an AI algebra; that is, $A$ is the $\mbox{C}^{*}$-algebra inductive limit of a sequence $$ A_{1}\stackrel{\phi_{1,2}}{\longrightarrow}A_{2}\stackrel{\phi_{2,3}}{\longrightarrow}A_{3} \longrightarrow\cdots\longrightarrow A_{n}\longrightarrow\cdots, $$ where $A_{n}=\bigoplus_{i=1}^{k_n}M_{[n,i]}(C(X^{i}_n))$, $X^{i}_n$ are $[0,1]$, $k_n$, and $[n,i]$ are positive integers. Suppose that $A$ has the ideal property: each closed two-sided ideal of $A$ is generated by the projections inside the ideal, as a closed two-sided ideal. In this article, we give a complete classification of AI algebras with the ideal property.

Keywords:AI algebras, K-group, tracial state, ideal property, classification
Categories:46L35, 19K14, 46L05, 46L08

7. CJM 2009 (vol 62 pp. 614)

Pronk, Dorette; Scull, Laura
Translation Groupoids and Orbifold Cohomology
We show that the bicategory of (representable) orbifolds and good maps is equivalent to the bicategory of orbifold translation groupoids and generalized equivariant maps, giving a mechanism for transferring results from equivariant homotopy theory to the orbifold category. As an application, we use this result to define orbifold versions of a couple of equivariant cohomology theories: K-theory and Bredon cohomology for certain coefficient diagrams.

Keywords:orbifolds, equivariant homotopy theory, translation groupoids, bicategories of fractions
Categories:57S15, 55N91, 19L47, 18D05, 18D35

8. CJM 2009 (vol 62 pp. 646)

Rupp, R.; Sasane, A.
Reducibility in AR(K), CR(K), and A(K)
Let $K$ denote a compact real symmetric subset of $\mathbb{C}$ and let $A_{\mathbb R}(K)$ denote the real Banach algebra of all real symmetric continuous functions on $K$ that are analytic in the interior $K^\circ$ of $K$, endowed with the supremum norm. We characterize all unimodular pairs $(f,g)$ in $A_{\mathbb R}(K)^2$ which are reducible. In addition, for an arbitrary compact $K$ in $\mathbb C$, we give a new proof (not relying on Banach algebra theory or elementary stable rank techniques) of the fact that the Bass stable rank of $A(K)$ is $1$. Finally, we also characterize all compact real symmetric sets $K$ such that $A_{\mathbb R}(K)$, respectively $C_{\mathbb R}(K)$, has Bass stable rank $1$.

Keywords:real Banach algebras, Bass stable rank, topological stable rank, reducibility
Categories:46J15, 19B10, 30H05, 93D15

9. CJM 2009 (vol 61 pp. 1073)

Griffiths, Ross; Lescop, Mikaël
On the $2$-Rank of the Hilbert Kernel of Number Fields
Let $E/F$ be a quadratic extension of number fields. In this paper, we show that the genus formula for Hilbert kernels, proved by M. Kolster and A. Movahhedi, gives the $2$-rank of the Hilbert kernel of $E$ provided that the $2$-primary Hilbert kernel of $F$ is trivial. However, since the original genus formula is not explicit enough in a very particular case, we first develop a refinement of this formula in order to employ it in the calculation of the $2$-rank of $E$ whenever $F$ is totally real with trivial $2$-primary Hilbert kernel. Finally, we apply our results to quadratic, bi-quadratic, and tri-quadratic fields which include a complete $2$-rank formula for the family of fields $\Q(\sqrt{2},\sqrt{\delta})$ where $\delta$ is a squarefree integer.

Categories:11R70, 19F15

10. CJM 2008 (vol 60 pp. 1387)

Romo, Fernando Pablos
On $n$-Dimensional Steinberg Symbols
The aim of this work is to provide a new approach for constructing $n$-dimensional Steinberg symbols on discrete valuation fields from $(n+1)$-cocycles and to study reciprocity laws on curves related to these symbols.

Keywords:Steinberg symbols, reciprocity laws, discrete valuation field, algebraic curves
Categories:19F15, 19D45, 19C09

11. CJM 2006 (vol 58 pp. 419)

Snaith, Victor P.
Stark's Conjecture and New Stickelberger Phenomena
We introduce a new conjecture concerning the construction of elements in the annihilator ideal associated to a Galois action on the higher-dimensional algebraic $K$-groups of rings of integers in number fields. Our conjecture is motivic in the sense that it involves the (transcendental) Borel regulator as well as being related to $l$-adic \'{e}tale cohomology. In addition, the conjecture generalises the well-known Coates--Sinnott conjecture. For example, for a totally real extension when $r = -2, -4, -6, \dotsc$ the Coates--Sinnott conjecture merely predicts that zero annihilates $K_{-2r}$ of the ring of $S$-integers while our conjecture predicts a non-trivial annihilator. By way of supporting evidence, we prove the corresponding (conjecturally equivalent) conjecture for the Galois action on the \'{e}tale cohomology of the cyclotomic extensions of the rationals.

Categories:11G55, 11R34, 11R42, 19F27

12. CJM 2005 (vol 57 pp. 225)

Booss-Bavnbek, Bernhelm; Lesch, Matthias; Phillips, John
Unbounded Fredholm Operators and Spectral Flow
We study the gap (= ``projection norm'' = ``graph distance'') topology of the space of all (not necessarily bounded) self-adjoint Fredholm operators in a separable Hilbert space by the Cayley transform and direct methods. In particular, we show the surprising result that this space is connected in contrast to the bounded case. Moreover, we present a rigorous definition of spectral flow of a path of such operators (actually alternative but mutually equivalent definitions) and prove the homotopy invariance. As an example, we discuss operator curves on manifolds with boundary.

Categories:58J30, 47A53, 19K56, 58J32

13. CJM 2005 (vol 57 pp. 180)

Somodi, Marius
On the Size of the Wild Set
To every pair of algebraic number fields with isomorphic Witt rings one can associate a number, called the {\it minimum number of wild primes}. Earlier investigations have established lower bounds for this number. In this paper an analysis is presented that expresses the minimum number of wild primes in terms of the number of wild dyadic primes. This formula not only gives immediate upper bounds, but can be considered to be an exact formula for the minimum number of wild primes.

Categories:11E12, 11E81, 19F15, 11R29

14. CJM 2004 (vol 56 pp. 926)

Hadfield, Tom
K-Homology of the Rotation Algebras $A_{\theta}$
We study the K-homology of the rotation algebras $A_{\theta}$ using the six-term cyclic sequence for the K-homology of a crossed product by ${\bf Z}$. In the case that $\theta$ is irrational, we use Pimsner and Voiculescu's work on AF-embeddings of the $A_{\theta}$ to search for the missing generator of the even K-homology.

Categories:58B34, 19K33, 46L

15. CJM 2001 (vol 53 pp. 1223)

Mygind, Jesper
Classification of Certain Simple $C^*$-Algebras with Torsion in $K_1$
We show that the Elliott invariant is a classifying invariant for the class of $C^*$-algebras that are simple unital infinite dimensional inductive limits of finite direct sums of building blocks of the form $$ \{f \in C(\T) \otimes M_n : f(x_i) \in M_{d_i}, i = 1,2,\dots,N\}, $$ where $x_1,x_2,\dots,x_N \in \T$, $d_1,d_2,\dots,d_N$ are integers dividing $n$, and $M_{d_i}$ is embedded unitally into $M_n$. Furthermore we prove existence and uniqueness theorems for $*$-homomorphisms between such algebras and we identify the range of the invariant.

Categories:46L80, 19K14, 46L05

16. CJM 2001 (vol 53 pp. 979)

Nagisa, Masaru; Osaka, Hiroyuki; Phillips, N. Christopher
Ranks of Algebras of Continuous $C^*$-Algebra Valued Functions
We prove a number of results about the stable and particularly the real ranks of tensor products of \ca s under the assumption that one of the factors is commutative. In particular, we prove the following: {\raggedright \begin{enumerate}[(5)] \item[(1)] If $X$ is any locally compact $\sm$-compact Hausdorff space and $A$ is any \ca, then\break $\RR \bigl( C_0 (X) \otimes A \bigr) \leq \dim (X) + \RR(A)$. \item[(2)] If $X$ is any locally compact Hausdorff space and $A$ is any \pisca, then $\RR \bigl( C_0 (X) \otimes A \bigr) \leq 1$. \item[(3)] $\RR \bigl( C ([0,1]) \otimes A \bigr) \geq 1$ for any nonzero \ca\ $A$, and $\sr \bigl( C ([0,1]^2) \otimes A \bigr) \geq 2$ for any unital \ca\ $A$. \item[(4)] If $A$ is a unital \ca\ such that $\RR(A) = 0$, $\sr (A) = 1$, and $K_1 (A) = 0$, then\break $\sr \bigl( C ([0,1]) \otimes A \bigr) = 1$. \item[(5)] There is a simple separable unital nuclear \ca\ $A$ such that $\RR(A) = 1$ and\break $\sr \bigl( C ([0,1]) \otimes A \bigr) = 1$. \end{enumerate}}

Categories:46L05, 46L52, 46L80, 19A13, 19B10

17. CJM 2001 (vol 53 pp. 631)

Walters, Samuel G.
K-Theory of Non-Commutative Spheres Arising from the Fourier Automorphism
For a dense $G_\delta$ set of real parameters $\theta$ in $[0,1]$ (containing the rationals) it is shown that the group $K_0 (A_\theta \rtimes_\sigma \mathbb{Z}_4)$ is isomorphic to $\mathbb{Z}^9$, where $A_\theta$ is the rotation C*-algebra generated by unitaries $U$, $V$ satisfying $VU = e^{2\pi i\theta} UV$ and $\sigma$ is the Fourier automorphism of $A_\theta$ defined by $\sigma(U) = V$, $\sigma(V) = U^{-1}$. More precisely, an explicit basis for $K_0$ consisting of nine canonical modules is given. (A slight generalization of this result is also obtained for certain separable continuous fields of unital C*-algebras over $[0,1]$.) The Connes Chern character $\ch \colon K_0 (A_\theta \rtimes_\sigma \mathbb{Z}_4) \to H^{\ev} (A_\theta \rtimes_\sigma \mathbb{Z}_4)^*$ is shown to be injective for a dense $G_\delta$ set of parameters $\theta$. The main computational tool in this paper is a group homomorphism $\vtr \colon K_0 (A_\theta \rtimes_\sigma \mathbb{Z}_4) \to \mathbb{R}^8 \times \mathbb{Z}$ obtained from the Connes Chern character by restricting the functionals in its codomain to a certain nine-dimensional subspace of $H^{\ev} (A_\theta \rtimes_\sigma \mathbb{Z}_4)$. The range of $\vtr$ is fully determined for each $\theta$. (We conjecture that this subspace is all of $H^{\ev}$.)

Keywords:C*-algebras, K-theory, automorphisms, rotation algebras, unbounded traces, Chern characters
Categories:46L80, 46L40, 19K14

18. CJM 2001 (vol 53 pp. 3)

Bell, J. P.
The Equivariant Grothendieck Groups of the Russell-Koras Threefolds
The Russell-Koras contractible threefolds are the smooth affine threefolds having a hyperbolic $\mathbb{C}^*$-action with quotient isomorphic to the corresponding quotient of the linear action on the tangent space at the unique fixed point. Koras and Russell gave a concrete description of all such threefolds and determined many interesting properties they possess. We use this description and these properties to compute the equivariant Grothendieck groups of these threefolds. In addition, we give certain equivariant invariants of these rings.

Categories:14J30, 19L47

19. CJM 2000 (vol 52 pp. 1310)

Yagunov, Serge
On the Homology of $\GL_n$ and Higher Pre-Bloch Groups
For every integer $n>1$ and infinite field $F$ we construct a spectral sequence converging to the homology of $\GL_n(F)$ relative to the group of monomial matrices $\GM_n(F)$. Some entries in $E^2$-terms of these spectral sequences may be interpreted as a natural generalization of the Bloch group to higher dimensions. These groups may be characterized as homology of $\GL_n$ relatively to $\GL_{n-1}$ and $\GM_n$. We apply the machinery developed to the investigation of stabilization maps in homology of General Linear Groups.

Categories:19D55, 20J06, 18G60

20. CJM 2000 (vol 52 pp. 47)

Chinburg, T.; Kolster, M.; Snaith, V. P.
Comparison of $K$-Theory Galois Module Structure Invariants
We prove that two, apparently different, class-group valued Galois module structure invariants associated to the algebraic $K$-groups of rings of algebraic integers coincide. This comparison result is particularly important in making explicit calculations.

Categories:11S99, 19F15, 19F27

21. CJM 1998 (vol 50 pp. 1048)

Goerss, P. G.; Jardine, J. F.
Localization theories for simplicial presheaves
Most extant localization theories for spaces, spectra and diagrams of such can be derived from a simple list of axioms which are verified in broad generality. Several new theories are introduced, including localizations for simplicial presheaves and presheaves of spectra at homology theories represented by presheaves of spectra, and a theory of localization along a geometric topos morphism. The $f$-localization concept has an analog for simplicial presheaves, and specializes to the $\hbox{\Bbbvii A}^1$-local theory of Morel-Voevodsky. This theory answers a question of Soul\'e concerning integral homology localizations for diagrams of spaces.

Categories:55P60, 19E08, 18F20

22. CJM 1998 (vol 50 pp. 673)

Carey, Alan; Phillips, John
Fredholm modules and spectral flow
An {\it odd unbounded\/} (respectively, $p$-{\it summable}) {\it Fredholm module\/} for a unital Banach $\ast$-algebra, $A$, is a pair $(H,D)$ where $A$ is represented on the Hilbert space, $H$, and $D$ is an unbounded self-adjoint operator on $H$ satisfying: \item{(1)} $(1+D^2)^{-1}$ is compact (respectively, $\Trace\bigl((1+D^2)^{-(p/2)}\bigr) <\infty$), and \item{(2)} $\{a\in A\mid [D,a]$ is bounded$\}$ is a dense $\ast-$subalgebra of $A$. If $u$ is a unitary in the dense $\ast-$subalgebra mentioned in (2) then $$ uDu^\ast=D+u[D,u^{\ast}]=D+B $$ where $B$ is a bounded self-adjoint operator. The path $$ D_t^u:=(1-t) D+tuDu^\ast=D+tB $$ is a ``continuous'' path of unbounded self-adjoint ``Fredholm'' operators. More precisely, we show that $$ F_t^u:=D_t^u \bigl(1+(D_t^u)^2\bigr)^{-{1\over 2}} $$ is a norm-continuous path of (bounded) self-adjoint Fredholm operators. The {\it spectral flow\/} of this path $\{F_t^u\}$ (or $\{ D_t^u\}$) is roughly speaking the net number of eigenvalues that pass through $0$ in the positive direction as $t$ runs from $0$ to $1$. This integer, $$ \sf(\{D_t^u\}):=\sf(\{F_t^u\}), $$ recovers the pairing of the $K$-homology class $[D]$ with the $K$-theory class [$u$]. We use I.~M.~Singer's idea (as did E.~Getzler in the $\theta$-summable case) to consider the operator $B$ as a parameter in the Banach manifold, $B_{\sa}(H)$, so that spectral flow can be exhibited as the integral of a closed $1$-form on this manifold. Now, for $B$ in our manifold, any $X\in T_B(B_{\sa}(H))$ is given by an $X$ in $B_{\sa}(H)$ as the derivative at $B$ along the curve $t\mapsto B+tX$ in the manifold. Then we show that for $m$ a sufficiently large half-integer: $$ \alpha (X)={1\over {\tilde {C}_m}}\Tr \Bigl(X\bigl(1+(D+B)^2\bigr)^{-m}\Bigr) $$ is a closed $1$-form. For any piecewise smooth path $\{D_t=D+B_t\}$ with $D_0$ and $D_1$ unitarily equivalent we show that $$ \sf(\{D_t\})={1\over {\tilde {C}_m}} \int_0^1\Tr \Bigl({d\over {dt}} (D_t)(1+D_t^2)^{-m}\Bigr)\,dt $$ the integral of the $1$-form $\alpha$. If $D_0$ and $D_1$ are not unitarily equivalent, we must add a pair of correction terms to the right-hand side. We also prove a bounded finitely summable version of the form: $$ \sf(\{F_t\})={1\over C_n}\int_0^1\Tr\Bigl({d\over dt}(F_t)(1-F_t^2)^n\Bigr)\,dt $$ for $n\geq{{p-1}\over 2}$ an integer. The unbounded case is proved by reducing to the bounded case via the map $D\mapsto F=D(1+D^2 )^{-{1\over 2}}$. We prove simultaneously a type II version of our results.

Categories:46L80, 19K33, 47A30, 47A55

23. CJM 1997 (vol 49 pp. 1265)

Snaith, V. P.
Hecke algebras and class-group invariant
Let $G$ be a finite group. To a set of subgroups of order two we associate a $\mod 2$ Hecke algebra and construct a homomorphism, $\psi$, from its units to the class-group of ${\bf Z}[G]$. We show that this homomorphism takes values in the subgroup, $D({\bf Z}[G])$. Alternative constructions of Chinburg invariants arising from the Galois module structure of higher-dimensional algebraic $K$-groups of rings of algebraic integers often differ by elements in the image of $\psi$. As an application we show that two such constructions coincide.

Categories:16S34, 19A99, 11R65

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