Andrew J. Dean, Department of Mathematical Sciences, Lakehead University, 955 Oliver Road, Thunder Bay, Ontario, Canada P7B 5E1; e-mail: ajdean@lakeheadu.ca

Sarah K. Lucky , Department of Mathematical Sciences, Lakehead University, 955 Oliver Road, Thunder Bay, Ontario, Canada P7B 5E1; e-mail: sklucky@lakeheadu.ca

Abstract/Résumé:

It is shown that if \(G\) is a lattice ordered countable group, then every action of \(({\Bbb Z}/2{\Bbb Z})\times ({\Bbb Z}/2{\Bbb Z})\) on \(G\) arises as an inductive limit of actions of \(({\Bbb Z}/2{\Bbb Z})\times ({\Bbb Z}/2{\Bbb Z})\) on simplicial groups. Some parts of the argument work in greater generality, and are proved for general finite abelian groups. A template is given for proving similar results for other such groups.

On montre que si \(G\) est un groupe dénombrable treillis-ordonné, alors toute action de \(({\Bbb Z}/2{\Bbb Z})\times ({\Bbb Z}/2{\Bbb Z})\) sur \(G\) provient d’une limite inductive d’actions de \(({\Bbb Z}/2{\Bbb Z})\times ({\Bbb Z}/2{\Bbb Z})\) sur des groupes simpliciaux. Des parties de cet argument fonctionnent dans une généralité plus grande et sont prouvées pour des groupes abéliens finis en général. Un modèle est donné pour prouver des résultats similaires pour d’autres groupes de ce type.

Keywords: Dimension groups, K-theory, classification

AMS Subject Classification: Classifications of $C^*$-algebras; factors, Noncommutative dynamical systems 46L35, 46L55

PDF(click to download): Actions of $({Z}/2{Z})$ x $({Z}/2{Z})$ on Lattice Ordered Dimension Groups

George A. Elliott, Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4; e-mail: elliott@math.toronto.edu

Qingzhai Fan , Department of Mathematics, Shanghai Maritime University, Shangha, China 201306; e-mail: fanqingzhai@fudan.edu.cn,qzfan@shmtu.edu.cn

Xiaochun Fang, Department of Mathematics, Tongji University, Shanghai, China 200092; e-mail: xfang@tongji.edu.cn

Abstract/Résumé:

In this paper, we introduce some classes of generalized tracial approximation C*-algebras. Consider the class of unital C*-algebras which are tracially 𝒵-absorbing (or have tracial nuclear dimension at most n, or have the property SP, or are m-almost divisible). Then A is tracially 𝒵-absorbing (respectively, has tracial nuclear dimension at most n, has the property SP, is weakly (n, m)-almost divisible) for any simple unital C*-algebra A in the corresponding class of generalized tracial approximation C*-algebras. As an application, let A be an infinite-dimensional unital simple C*-algebra, and let B be a centrally large subalgebra of A. If B is tracially 𝒵-absorbing, then A is tracially 𝒵-absorbing. This result was obtained by Archey, Buck, and Phillips in Archey et al. (2018).

On introduit la notion d’approximation traciale généralisée d’une C*-algèbre par des C*-algèbres dans une class donnée. Cette notion généralise la notion de Lin d’approximation triviale simple, et aussi la notion d’Archey et de Phillips de centralement grande sousalgèbre, deux notions qui se sont démontrées très importantes.

Keywords: Cuntz Semigroup, C∗-algebras, tracial approximation

AMS Subject Classification: General theory of $C^*$-algebras, Classifications of $C^*$-algebras; factors, K-theory and operator algebras -including cyclic theory 46L05, 46L35, 46L80

PDF(click to download): Generalized Tracially Approximated C*-algebras

George A. Elliott, Department of Mathematics, University of Toronto, Toronto, Canada M5S 2E4; e-mail: elliott@math.toronto.edu

Abstract/Résumé:

It is shown that for a unital C*-algebra, what is sometimes referred to as the Elliott invariant—loosely speaking, K-theory and traces— i.e., the order-unit K\(_0\)-group, the K\(_1\)-group, and the trace simplex, paired in the natural way with K\(_0\), can be expressed purely in terms of K-theory, with the trace simplex and its pairing with K\(_0\) recoverable in a simple way (using polar decomposition) from algebraic K\(_1\), defined as in the purely algebraic context using invertible elements rather than just unitaries.

L’invariant naïf d’Elliott, qui est à la base de la classification complète récente d’une énorme classe de C*-algèbres simples (celles qui sont de dimension nucléaire finie, qui sont séparables, et qui satisfont à l’UCT), peut s’exprimer entièrement dans le cadre de K-théorie algébrique.

Keywords: Algebraic K1-group of a C*-algebra encodes bounded traces

AMS Subject Classification: None of the above; but in this section, $K_0$ as an ordered group; traces, General theory of $C^*$-algebras, Classifications of $C^*$-algebras; factors, K-theory and operator algebras -including cyclic theory 19B99, 19K14, 46L05, 46L35, 46L80

PDF(click to download): K-Theory and Traces

Guihua Gong, Department of Mathematics, Hebei Normal University, Shijiazhuang, Hebei 050016, China
and Department of Mathematics, University of Puerto Rico, Rio Piedras, PR 00936, USA; e-mail: ghgong@gmail.com

Huaxin Lin, Department of Mathematics, East China Normal University, Shanghai 200062, China and
(Current) Department of Mathematics, University of Oregon, Eugene, Oregon, 97402, USA; e-mail: hlin@uoregon.edu

Zhuang Niu, Department of Mathematics, University of Wyoming, Laramie, WY, USA, 82071; e-mail: zniu@uwyo.edu,

Abstract/Résumé:

A classification theorem is obtained for a class of unital simple separable amenable \({\cal Z}\)-stable C*-algebras which exhausts all possible values of the Elliott invariant for unital stably finite simple separable amenable \({\cal Z}\)-stable C*-algebras. Moreover, it contains all unital simple separable amenable C*-algebras which satisfy the UCT and have finite rational tracial rank.

Dans cet article et le précédent on donne une classification complète, au moyen de l’invariant d’Elliott, d’une sous-classe de la classe des C*-algèbres simples, moyennables, séparables, à élément unité, absorbant l’algèbre de Jiang-Su, et satisfaisant au UCT, qui épuise l’ensemble des valeurs possibles de l’invariant pour cette class. La partie I réalise une grande partie de ce projet, et la partie II l’achève.

Keywords: Classication of Simple C*-algebras

AMS Subject Classification: General theory of $C^*$-algebras, Classifications of $C^*$-algebras; factors, K-theory and operator algebras -including cyclic theory 46L05, 46L35, 46L80

PDF(click to download): A Classification of Finite Simple Amenable Z-stable C*-algebras, II: C*-algebras with Rational Generalized Tracial Rank One

Guihua Gong, Department of Mathematics, Hebei Normal University, Shijiazhuang, Hebei 050016, China and Department of Mathematics, University of Puerto Rico, Rio Piedras, PR 00936, USA; e-mail: ghgong@gmail.com

Huaxin Lin, Department of Mathematics, East China Normal University, Shanghai 200062, China and (Current) Department of Mathematics, University of Oregon, Eugene, Oregon, 97402, USA; e-mail: hlin@uoregon.edu

Zhuang Niu, Department of Mathematics, University of Wyoming, Laramie, WY, USA, 82071; e-mail: zniu@uwyo.edu

Abstract/Résumé:

A class of C*-algebras, to be called those of generalized tracial rank one, is introduced. A second class of unital simple separable amenable C*-algebras, those whose tensor products with UHF-algebras of infinite type are in the first class, to be referred to as those of rational generalized tracial rank one, is proved to exhaust all possible values of the Elliott invariant for unital finite simple separable amenable \({\cal Z}\)-stable C*-algebras. A number of results toward the classification of the second class are presented including an isomorphism theorem for a special sub-class of the first class, leading to the general classification of all unital simple s with rational generalized tracial rank one in Part II.

Dans cet article et le prochain, on donne une classification complète, au moyen de l’invariant d’Elliott, d’une sous-classe de la classe des C*-algèbres simples, moyennables, séparables, à élément unité, absorbant l’algèbre de Jiang-Su, et satisfaisant au UCT, qui épuise l’ensemble des valeurs possibles de l’invariant pour cette class. La partie I réalise une grande partie de ce projet, et la partie II l’achève.

Keywords: Classification of simple C*-algebras

AMS Subject Classification: General theory of $C^*$-algebras, Classifications of $C^*$-algebras; factors, K-theory and operator algebras -including cyclic theory 46L05, 46L35, 46L80

PDF(click to download): A Classification of Finite Simple Amenable Z-stable C*-algebras, I: C*-algebras with Generalized Tracial Rank One

George A. Elliott, FRSC,Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4; e-mail: elliott@math.toronto.edu,

Qingzhai Fan,Department of Mathematics, Shanghai Maritime University, Shanghai, China 201306; e-mail: fanqingzhai@fudan.edu.cn,qzfan@shmtu.edu.cn

Xiaochun Fang,Department of Mathematics, Tongji University, Shangha, China 200092; e-mail: xfang@mail.tongji.edu.cn

Abstract/Résumé:

We show that the following properties of the \({\rm C^*}\)-algebras in a class \(\Omega\) are inherited by simple unital \({\rm C^*}\)-algebras in the class \({\rm TA}\Omega\): \((1)\) \(\beta\)-comparison (\(1\leq \beta < \infty\)), \((2)\) \(n\)-comparison, \((3)\) trace \(\mathcal{Z}\)– absorption, \((4)\) \(m\)-almost divisibility, \((5)\) \((n,m) ~(m\neq 0)\) comparison, and \((6)\) tracial approximate divisibility. As an application, every unital simple \({\rm C^*}\)-algebra with tracial topological rank at most \(k\) has the property of \(k\)-comparison. Also as an application, let \(A\) be an infinite-dimensional simple unital \({\rm C^*}\)-algebra such that \(A\) has one of the above-listed properties. Suppose that \(\alpha: G\to {\rm Aut}(A)\) is an action of a finite group \(G\) on \(A\) which has the tracial Rokhlin property. Then the crossed product \({\rm C^*}\)-algebra \({\rm C^*}( G, A,\alpha)\) also has the property under consideration.

On considère plusieurs propriétés d’une C*-algèbre simple à élément unité qui sont héritées par approximation traciale. Comme application on démontre que ces propriétés sont aussi héritées par la C*-algèbre produit croisé associée à une action d’un groupe fini qui possède la propriété de Rokhlin traciale.

Keywords: C*-algebra, Cuntz Semigroup, tracial approximation

AMS Subject Classification: General theory of $C^*$-algebras, Classifications of $C^*$-algebras; factors, K-theory and operator algebras -including cyclic theory 46L05, 46L35, 46L80

PDF(click to download): Certain Properties of Tracial Approximation ${C^*}$-Algebras

Sam Walters,Department of Mathematics and Statistics, University of Northern British Columbia, Prince
George, BC, V2N 4Z9, Canada; e-mail: walters@unbc.ca

Abstract/Résumé:

We prove that for locally compact, compactly generated self-dual Abelian groups \(G\), there are canonical unitary integral operators on \(L^2(G)\) analogous to the Fourier transform but which have orders 3 and 6. To do this, we establish the existence of a certain projective character on \(G\) whose phase multiplication with the FT gives rise to the Cubic transform (of order 3). (Thus, although the Fourier transform has order 4, one can “make it” have order 3 (or 6) by means of a phase factor!)

Soit \(G\) un groupe localement compact, engendré par un sousensemble compact, et isomorphe à son groupe dual. On construit des operateurs intégrals unitaires canoniques qui sont analogues à la transformée de Fourier, mais qui sont d’ordres trois et six.

Keywords: C*-algebra, Fourier transform, Gaussian sums, Hilbert space, L2 spaces, Locally compact Abelian groups, characters, cyclic groups, integral transforms, projective char- acter, self-dual groups, unitary operators

AMS Subject Classification: Harmonic analysis and almost periodicity, General properties and structure of LCA groups, Compact groups, General properties and structure of locally compact groups, $C^*$-algebras and $W$*-algebras in relation to group representations, General properties and structure of real Lie groups, Integral representations; integral operators; integral equations methods, Integral operators, Classifications of $C^*$-algebras; factors, Automorphisms, K-theory and operator algebras -including cyclic theory 11K70, 22B05, 22C05, 22D05, 22D25, 22E15, 31A10, 45P05, 46L35, 46L40, 46L80

PDF(click to download): Cubic and Hexic Integral Transforms for Locally Compact Abelian Groups

Sam Walters,Department of Mathematics and Statistics, University of Northern British Columbia, Prince
George, BC V2N 4Z9, Canada; e-mail: walters@unbc.ca

Abstract/Résumé:

We report on recent results on the existence of Cubic and Hexic integral transforms on self-dual locally compact groups (orders 3 and 6 analogues of the classical Fourier transform) and their application in constructing a canonical continuous section of smooth projections \(\mathcal E(t)\) of the continuous field of rotation C*-algebras \(\{A_t\}_{0 \le t \le 1}\) that is invariant under the noncommutative Hexic transform automorphism. This leads to invariant matrix (point) projections of the irrational noncommutative tori \(A_\theta\). We also present a quick method for computing the (quantized) topological invariants of such projections using techniques from classical Theta function theory.

On décrit des résultats récents sur l’existence d’une transformation intégrale d’ordre trois (ou d’ordre six) sur un groupe localement compact abélien self-dual. On étudie l’application possible à la construction d’un champs continu de projecteurs invariants sous l’automorphisme associé du champs de C*-algèbres de rotation. On calcule certains invariants topologiques de ces projecteurs.

Keywords: C*-algebra, Fourier transform, automorphisms, inductive limits, noncommutative tori, orbifold, rotation algebra, symmetries, topological invariants, unbounded traces

AMS Subject Classification: Classifications of $C^*$-algebras; factors, Automorphisms, K-theory and operator algebras -including cyclic theory, $K$-theory, String and superstring theories; other extended objects (e.g.; branes), Topological field theories, String and superstring theories 46L35, 46L40, 46L80, 55N15, 81T30, 81T45, 83E30

PDF(click to download): Periodic Integral Transforms and Associated Noncommutative Orbifold Projections

Sam Walters,Department of Mathematics and Statistics, University of Northern British Columbia, Prince
George, BC, V2N 4Z9, Canada; e-mail: walters@unbc.ca

Abstract/Résumé:

We demonstrate, in a rather quantitative manner, the existence of topological obstructions to approximating the irrational rotation C*-algebra \(A_\theta\) by Fourier invariant unital C*-subalgebras of either of the forms \[M \oplus B \oplus \sigma(B), \qquad M \oplus N \oplus D \oplus \sigma(D) \oplus \sigma^2(D) \oplus \sigma^3(D),\] where \(M, N\) are Fourier invariant matrix algebras (over \(\mathbb C\)), \(B\) is a C*-subalgebra whose unit projection is flip invariant and orthogonal to its Fourier transform, and \(D\) is a C*-subalgebra whose unit projection is orthogonal to its orbit under the Fourier transform. Here, \(\sigma\) is the noncommutative Fourier transform automorphism of \(A_\theta\) defined by \(\sigma(U) = V^{-1},\ \sigma(V)=U\) on the canonical unitary generators \(U,V\) obeying the unitary Heisenberg commutation relation \(VU = e^{2\pi i\theta}UV\).

On montre l’existence d’obstructions topologiques à l’approximation du tore non-commutatif par sous-algèbres de certains types qui sont invariantes sous l’automorphisme de Fourier.

Keywords: C*-algebra, Fourier transform, automorphisms, inductive limits, noncommutative tori, rotation algebra, topological invariants, topological obstructions, unbounded traces

AMS Subject Classification: Classifications of $C^*$-algebras; factors, Automorphisms, K-theory and operator algebras -including cyclic theory 46L35, 46L40, 46L80

PDF(click to download): Topological Obstruction to Approximating the Irrational Rotation C*-algebra by Certain Fourier Invariant C*-subalgebras

Zhuang Niu,Department of Mathematics, University of Wyoming, Laramie, Wyoming, 82071 USA; e-mail: zniu@uwyo.edu

Abstract/Résumé:

Motivated by Huaxin Lin’s axiomatization of AH-algebras, the class of TASI-algebras is introduced as the class of unital C*-algebras which can be tracially approximated by splitting interval algebras—certain sub-C*-algebras of interval algebras. It is shown that the class of simple separable nuclear TASI-algebras satisfying the UCT is classified by the Elliott invariant. As a consequence, any such TASI-algebra is then isomorphic to an inductive limit of splitting interval algebras together with certain homogeneous C*-algebras—so it is an ASH-algebra.

Une classe de C*-algèbres qui généralisent la classe bien connue TAI de Lin est considérée—basées sur, au lieu de l’intervalle, ce qui pourrait s’appeler l’intervalle fendu (“splitting interval”), de sorte que l’on les appelle la classe TASI. On montre que la classe de C*-algèbres TASI qui sont simples, nucléaires, et à élément unité, qui vérifient le théorème à coefficients universels (UCT), peuvent se classifier d’après l’invariant d’Elliott.

Keywords: Classification of simple C*-algebras, inductive limits of sub-homogeneous C*- algebras, tracially approximate splitting interval algebras

AMS Subject Classification: Classifications of $C^*$-algebras; factors 46L35

PDF(click to download): A Classification of Tracially Approximate Splitting Interval Algebras. III. Uniqueness Theorem and Isomorphism Theorem

Zhuang Niu, Department of Mathematics, University of Wyoming, Laramie, Wyoming, 82071 USA; e-mail: zniu@uwyo.edu

Abstract/Résumé:

Motivated by Huaxin Lin’s axiomatization of AH-algebras, the class of TASI-algebras is introduced as the class of unital C*-algebras which can be tracially approximated by splitting interval algebras—certain sub-C*-algebras of interval algebras. It is shown that the class of simple separable nuclear TASI-algebras satisfying the UCT is classified by the Elliott invariant. As a consequence, any such TASI-algebra is then isomorphic to an inductive limit of splitting interval algebras together with certain homogeneous C*-algebras—so it is an ASH-algebra.

Une classe de C*-algèbres qui généralisent la classe bien connue TAI de Lin est considérée—basées sur, au lieu de l’intervalle, ce qui pourrait s’appeler l’intervalle fendu ("splitting interval"), de sorte que l’on les appelle la classe TASI. On montre que la classe de C*-algèbres TASI qui sont simples, nucléaires, et à élément unité, qui vérifient le théorème à coefficients universels (UCT), peuvent se classifier d’après l’invariant d’Elliott.

Keywords: Classification of simple C*-algebras, inductive limits of sub-homogeneous C*- algebras, tracially approximate splitting interval algebras

AMS Subject Classification: Classifications of $C^*$-algebras; factors 46L35

PDF(click to download): A Classification of Tracially Approximate Splitting Interval Algebras. II. Existence Theorem

Zhuang Niu, Department of Mathematics, University of Wyoming, Laramie, Wyoming, USA 82071; e-mail: zniu@uwyo.edu

Abstract/Résumé:

Motivated by Huaxin Lin’s axiomatization of AH-algebras, the class of TASI-algebras is introduced as the class of unital C*-algebras which can be tracially approximated by splitting interval algebras—certain sub-C*-algebras of interval algebras. It is shown that the class of simple separable nuclear TASI-algebras satisfying the UCT is classified by the Elliott invariant. As a consequence, any such TASI-algebra is then isomorphic to an inductive limit of splitting interval algebras together with certain homogeneous C*-algebras—so it is an ASH-algebra.

Une classe de C*-algèbres qui généralisent la classe bien connue TAI de Lin est considérée—basées sur, au lieu de l’intervalle, ce qui pourrait s’appeler l’intervalle fendu (“splitting interval”), de sorte que l’on les appelle la classe TASI. On montre que la classe de C*-algèbres TASI qui sont simples, nucléaires, et à élément unité, qui vérifient le théorème à coefficients universels (UCT), peuvent se classifier d’après l’invariant d’Elliott.

Keywords: Classification of simple C*-algebras, inductive limits of sub-homogeneous C*- algebras, tracially approximate splitting interval algebras

AMS Subject Classification: Classifications of $C^*$-algebras; factors 46L35

PDF(click to download): A Classification of Tracially Approximate Splitting Interval Algebras~~I. The Building Blocks and the Limit Algebras

Andrew S. Toms, Department of Mathematics, Purdue University, 150 N. University St., West Lafayette, IN 47907, USA; e-mail: atoms@purdue.edu

Abstract/Résumé:

We observe almost divisibility for the original Cuntz semigroup of a simple AH algebra with strict comparison. As a consequence, the properties of strict comparison, finite nuclear dimension, and \(\mathcal{Z}\)-stability are equivalent for such algebras, confirming partially a conjecture of Winter and the author.

\(\mathcal{Z}\)-stabilité sont équivalentes pour une telle algèbre, ce qui confirme partiellement une conjecture de Winter et de l’auteur.

Keywords:

AMS Subject Classification: Classifications of $C^*$-algebras; factors 46L35

PDF(click to download): Characterizing classifiable AH algebras

Lawrence G. Brown, Department of Mathematics, Purdue University, West Lafayette, IN 47907–2067 USA; email: lgb@math.purdue.edu

Abstract/Résumé:

We show that if \(A\) is a \(C^*\)-algebra of real rank zero and stable rank one, then the Riesz interpolation property holds in the ordered group \(K_0(A) \oplus K_1(A)\).

Nous montrons que si \(A\) est une \(C^*\)-algèbre de rang réel zéro et de rang stable égal à un, donc la propriété d’interpolation de Riesz est valable dans le groupe ordonné \(K_0(A) \oplus K_1(A)\).

Keywords:

AMS Subject Classification: Classifications of $C^*$-algebras; factors 46L35

PDF(click to download): The Riesz interpolation property for $K_0(A) oplus K_1(A)$

Z. Niu

Abstract/Résumé:

No abstract available but the full text pdf may be downloaded at the title link below.

Keywords:

AMS Subject Classification: Classifications of $C^*$-algebras; factors, K-theory and operator algebras -including cyclic theory 46L35, 46L80

PDF(click to download): On the Classification of TAI algebras

E.J. Beggs / P. Goldstein

Abstract/Résumé:

No abstract available but the full text pdf may be downloaded at the title link below.

Keywords:

AMS Subject Classification: Tensor products of $C^*$-algebras, Classifications of $C^*$-algebras; factors, Noncommutative dynamical systems 46L06, 46L35, 46L55

PDF(click to download): Maximal abelian subalgebras of ?n

G.A. Elliott / T. Giordano

Abstract/Résumé:

No abstract available but the full text pdf may be downloaded at the title link below.

Keywords:

AMS Subject Classification: None of the above; but in this section, Classifications of $C^*$-algebras; factors, 28D99, 46L35, 60J15

PDF(click to download): Every approximately transitive amenable action of a locally compact group is a Poisson boundary