Chris Bruce, School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8QQ, United Kingdom; e-mail: Chris.Bruce@glasgow.ac.uk

Charles Starling, Carleton University, School of Mathematics and Statistics, 4302 Herzberg Laboratories, Ottawa, ON K1S 5B6; e-mail: cstar@math.carleton.ca

Abstract/Résumé:

We correct the proof of Theorem 4.1 from [C. R. Math. Acad. Sci. Soc. R. Can. 44 (2022), no. 4, 88–112].

Nous corrigeons la démonstration du théorème 4.1 dans l’article [C. R. Math. Acad. Sci. Soc. R. Can. 44 (2022), no. 4, 88–112].

Keywords: C*-algebra, LCM semigroup, groupoid, inverse semigroup, subshift, tight representation, uniqueness

AMS Subject Classification: Groupoids; semigroupoids; semigroups; groups (viewed as categories), Inverse semigroups, General theory of $C^*$-algebras 18B40, 20M18, 46L05

PDF(click to download): Corrigendum to ``A New Uniqueness Theorem for the Tight C*-algebra of an Inverse Semigroup'' [C. R. Math. Acad. Sci. Soc. R. Can. 44 (2022), no. 4, 88--112]

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

Mohammad Rouzbehani, School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran; e-mail: rouzbehani.m.math@gmail.com

Abstract/Résumé:

We show that a C*-algebra with topological dimension zero is purely infinite if it is weakly purely infinite (a question of Kirchberg and Rørdam). We give an application of this result.

On démontre qu’une C*-algèbre de dimension topologique égale à zéro est purement infinie si elle est faiblement purement infinie (une question de Kirchberg et de Rørdam). On donne une application de ce résultat.

Keywords: (weak) pure infiniteness, C*-algebra, topological dimension zero

AMS Subject Classification: General theory of $C^*$-algebras 46L05

PDF(click to download): Weakly Purely Infinite C*-algebras with Topological Dimension Zero are Purely Infinite

Charles Starling, Carleton University, School of Mathematics and Statistics. 4302 Herzberg Laboratories, Ottawa ON, K1S 5B6; e-mail: cstar@math.carleton.ca

Abstract/Résumé:

We prove a new uniqueness theorem for the tight C*-algebras of an inverse semigroup by generalising the uniqueness theorem given for étale groupoid C*-algebras by Brown, Nagy, Reznikoff, Sims, and Williams. We use this to show that in the nuclear and Hausdorff case, a *-homomorphism from the boundary quotient C*-algebra of a right LCM monoid is injective if and only if it is injective on the subalgebra generated by the core submonoid. We also use our result to clarify the identity of the tight C*-algebra of an inverse semigroup we previously associated to a subshift and erroneously identified as the Carlsen-Matsumoto algebra.

Nous prouvons un nouveau thèoréme d’unicité pour les C*-algèbres serrées d’un semi-groupe inverse en généralisant le théorème d’unicité donné pour les C*-algèbres groupoides étales par Brown, Nagy, Reznikoff, Sims et Williams. Nous utilisons ceci pour montrer que dans le cas nucléaire et de Hausdorff, un *-homomorphisme de l’algèbre C* du quotient aux limites d’un monoïde LCM droit est injectif si et seulement s’il est injectif sur la sous-algèbre générée par le sous-monoide de noyau. Nous utilisons également notre résultat pour clarifier l’identité de l’algèbre C* serrée d’un semi-groupe inverse que nous avons précédemment associé à un sous-décalage et identifié à tort comme l’algèbre de Carlsen-Matsumoto.

Keywords: C*-algebra, LCM semigroup, groupoid, inverse semigroup, subshift, tight representation, uniqueness

AMS Subject Classification: Groupoids; semigroupoids; semigroups; groups (viewed as categories), Inverse semigroups, General theory of $C^*$-algebras 18B40, 20M18, 46L05

PDF(click to download): A New Uniqueness Theorem for the Tight C*-algebra of an Inverse Semigroup

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

Gilles G. de Castro, Departamento de Matemática, Universidade Federal de Santa Catarina, 88040-970 Florianópolis SC, Brazil; e-mail: gilles.castro@ufsc.br

Fernando de L. Mortari, Departamento de Matemática, Universidade Federal de Santa Catarina, 88040-970 Florianópolis SC, Brazil; e-mail: fernando.mortari@ufsc.br

Abstract/Résumé:

Given a positive function on the set of edges of an arbitrary directed graph \(E=(E^0,E^1)\), we define a one-parameter group of automorphisms on the C*-algebra of the graph \(C^*(E)\), and study the problem of finding KMS states for this action. We prove that there are bijective correspondences between KMS states on \(C^*(E)\), a certain class of states on its core, and a certain class of tracial states on \(C_0(E^0)\). We also find the ground states for this action and give some examples.

Étant donné une fonction positive sur l’ensemble des arcs d’un graphe orienté arbitraire \(E=(E^0,E^1)\), nous définissons un groupe à un paramètre d’automorphismes de la \(C^*\)-algèbre du graphe \(C^*(E)\), et nous étudions le problème de trouver les états KMS pour cette action. Nous prouvons qu’il existe des bijections entre les états KMS sur \(C^*(E)\), une certaine classe d’états sur le core, et une certaine classe détats traciaux sur \(C_0(E^0)\). Nous trouvons également les états fondamentaux pour cette action et nous donnons quelques exemples.

Keywords: C*-algebra, Graph, Ground state, KMS state

AMS Subject Classification: Noncommutative dynamical systems 46L55

PDF(click to download): KMS States for the Generalized Gauge Action on Graph Algebras

Dan Z. Kučerovský, Department of Mathematics and Statistics, University of New Brunswick, Federicton, NB, Canada E3B 5A3; e-mail: dkucerov@unb.ca

Abstract/Résumé:

We give a complete invariant for finite-dimensional Hopf C*-algebras. Algebras that are equal under the invariant are the same up to a Hopf *-(co-anti)isomorphism.

On donne un invariant complet pour les C*-algèbres de Hopf de dimension finie.

Keywords: C*-algebra, Hopf algebras

AMS Subject Classification: , Algebras of specific types of operators (Toeplitz; integral; pseudodifferential; etc.) 16T05, 47L80

PDF(click to download): On an Abstract Classification of Finite-dimensional Hopf C*-algebras

R. Exel, Departamento de Matem ́atica, Universidade Federal de Santa Catarina, 88040-970 Florian ́opolis SC, Brazil; e-mail: exel@mtm.ufsc.br

Abstract/Résumé:

Given two algebras $A$ and $B$, sometimes assumed to be C\*-algebras, we consider the question of putting algebra or C\*-algebra structures on the tensor product $A\otimes B$. In the C\*-case, assuming $B$ to be two-dimensonal, we characterize all possible such C\*-algebra structures in terms of an action of the cyclic group ${\mathbb Z}_2$. An example related to commuting squares is also discussed.

Si $A$ et $B$ sont deux algèbres (resp. deux C\*-algèbres), nous étudions dans cette note les structures possibles d’algèbre (resp. de C\*-algèbre) qui peuvent être définies sur le produit tensoriel $A\otimes B$. Si $A$ est une C\*-algèbre, nous caractérisons toutes les structures de C\*-algèbre sur le produit tensoriel $A\otimes \mathbb{C}^2$ par une action du groupe cyclique $\mathbb{Z}_2$. Nous présentons aussi un exemple associé aux carrés commutatifs.

Keywords: Algebra, C*-algebra, Jones’ basic construction, algebra structure, alloy, blend, commuting square, conditional expectation, crossed product, index finite type, tensor product

AMS Subject Classification: None of the above; but in this section, 16S99, 46L04

PDF(click to download): Blends and Alloys

Sandro Molina-Cabrera, Department of Mathematics, Rıo Piedras Campus, University of Puerto Rico, Box 23355, San Juan, Puerto Rico 00931-3355, USA; email: smolinacabrera@gmail.com

Abstract/Résumé:

We show that the \(K_0\)-group of an inductive limit of recursive subhomogeneous algebras with compact metrizable spaces of dimension at most one as local spectra is torsion free. This result implies that the \(K_0\)-group of a unital simple AH algebra which is the inductive limit of recursive subhomogeneous algebras, with compact metrizable spaces of dimension at most one as local spectra, is torsion free. This proves that Li’s reduction theorem for the dimension of the local spectra of unital simple AH algebras cannot be improved, in other words, that the dimension of the local spectra of unital simple AH algebras cannot be further reduced from two to one, even when we use subhomogeneous algebras. This also shows that if a reduction theorem for the dimension of the local spectra of simple inductive limits of recursive subhomogeneous algebras exists, then, after the reduction, the local spectra of the building blocks cannot always be one dimensional.

Nous démontrons que le \(K_0\)-groupe d’une limite inductive des algèbres sous-homogènes récursives, dont les spectres locaux consistent en des espaces compacts métrisables de dimension au plus un, n’a pas de torsion. Ce résultat implique que les \(K_0\)-groupes d’une algèbre AH simple et avec l’unité qui est la limite des algèbres sous-homogènes rećursives, dont les spectres locaux consistent en des espaces compacts métrisables de dimension au plus un, n’a pas de torsion. Cela prouve que le théorème de Li de la réduction pour la dimension des spectres locaux des algèbres AH simples et avec l’unité ne peut pas être améliorée, en d’autres termes, que la dimension des spectres locaux des algèbres AH simples et avec l’unité ne peut pas encore être réduit de deux à un, même quand on utilise des algèbres sous-homogènes. Cela montre aussi que si un théorème de réduction pour la dimension des spectres locaux d’une limite inductive simple des algèbres sous-homogènes récursives existe, alors, après la réduction, les spectres locaux des blocs de construction ne peuvent pas être toujours de dimension un.

Keywords: AH algebras, C*-algebra, K-theory, classification, recursive subhomogeneous algebras

AMS Subject Classification: K-theory and operator algebras -including cyclic theory 46L80

PDF(click to download): Torsion in the ${K_0}$-Group of a Recursive Subhomogeneous Algebra

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

Abstract/Résumé:

A remark on orthogonality of elements of a C*-algebra Resume/Abstract: It is shown that any two non-zero hereditary sub-C*-algebras of a C*-algebra that has no minimal projections have approximately orthogonal elements of norm one. (The question of exact orthogonality is left open.)

On démontre que, dans une C*-algèbre sans projecteur minimal, deux sous-C*-algèbres héréditaires qui ne sont pas égales à zéro possèdent des éléments de norme un qui sont approximativement orthogonals.

Keywords: C*-algebra, approximate orthogonality, hereditary subalgebras, orthogonality

AMS Subject Classification: General theory of $C^*$-algebras 46L05

PDF(click to download): A remark on orthogonality of elements of a C*-algebra

Hiroyuki Osaka, Department of Mathematical Sciences, Ritsumeikan University, Kusatsu, Shiga, 525-8577 Japan; email: osaka@se.ritsumei.ac.jp

Tamotsu Teruya, College of Science and Engineering, Ritsumeikan University, Kusatsu, Shiga, 525-8577 Japan; email: teruya@se.ritsumei.ac.jp

Abstract/Résumé:

Let \(1 \in A \subset B\) be an inclusion of unital \(C^*\)-algebras of index-finite type and depth \(2\). Suppose that \(A\) is infinite dimensional, simple, with the property \(\operatorname{SP}\). We prove that if \(\operatorname{tsr}(A) = 1\), then \(\operatorname{tsr}(B) \leq 2\). An interesting special case is \(B = A \rtimes_\alpha G\), where \(\alpha\) is an action of a finite group \(G\) on \(\operatorname{Aut}(A)\).

Soit \(1 \in A \subset B\) une inclusion de \(C^*\)-algèbres unitals du type indice-fini et de profondeur \(2\). On suppose que \(A\) est de dimension infinie, simple, et que \(A\) a la propriété \(\operatorname{SP}\). On démontre que, si \(\operatorname{tsr}(A) = 1\), donc \(\operatorname{tsr}(B) \leq 2\). Un cas intéressant est \(B = A \rtimes_\alpha G\), oú \(\alpha\) est une action d’un groupe fini \(G\) sur \(\operatorname{Aut}(A)\).

Keywords: C*-algebra, property SP, stable rank

AMS Subject Classification: General theory of $C^*$-algebras 46L05

PDF(click to download): Stable rank of depth two inclusions of $C^*$-algebras

S. Walters

Abstract/Résumé:

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

Keywords: C*-algebra, Fourier transform, automorphisms, integral transforms, rotation algebras

AMS Subject Classification: General transforms, Special transforms (Legendre; Hilbert; etc.), Automorphisms, K-theory and operator algebras -including cyclic theory 44A05, 44A15, 46L40, 46L80

PDF(click to download): Periodic Integral Transforms and C*-Algebras

A. El Kinani / A. Ifzarne

Abstract/Résumé:

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

Keywords: C*-algebra, algèbre hermitienne, algèbre p-Banach, anti-morphisme involutif

AMS Subject Classification: Automorphisms and endomorphisms, General theory of topological algebras with involution 16W20, 46K05

PDF(click to download): Anti-morphisme involutif et algèbres p-Banach hermitiennes

S.G. Walters

Abstract/Résumé:

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

Keywords: C*-algebra, Chern characters, K-theory, automorphisms, rotation algebras, unbounded traces

AMS Subject Classification: $K_0$ as an ordered group; traces, Automorphisms, K-theory and operator algebras -including cyclic theory 19K14, 46L40, 46L80

PDF(click to download): On the irrational quartic algebra

J. Phillips/ I. Raeburn

Abstract/Résumé:

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

Keywords: C*-algebra

AMS Subject Classification:

PDF(click to download): Automorphisms of C*-algebras and second Cech cohomology

John Phillips / Iain Rathburn

Abstract/Résumé:

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

Keywords: C*-algebra

AMS Subject Classification:

PDF(click to download): Perturbations of C*-Algebras