scholarly journals Deformations of Fell bundles and twisted graph algebras

2016 ◽  
Vol 161 (3) ◽  
pp. 535-558 ◽  
Author(s):  
IAIN RAEBURN
Keyword(s):  
Discrete Groups ◽  
Graph Algebras ◽  
Graph Algebra

AbstractWe consider Fell bundles over discrete groups, and theC*-algebra which is universal for representations of the bundle. We define deformations of Fell bundles, which are new Fell bundles with the same underlying Banach bundle but with the multiplication deformed by a two-cocycle on the group. Every graph algebra can be viewed as theC*-algebra of a Fell bundle, and there are many cocycles of interest with which to deform them. We thus obtain many of the twisted graph algebras of Kumjian, Pask and Sims. We demonstate the utility of our approach to these twisted graph algebras by proving that the deformations associated to different cocycles can be assembled as the fibres of aC*-bundle.

scholarly journals Graph algebras and orbit equivalence

2015 ◽  
Vol 37 (2) ◽  
pp. 389-417 ◽  
Author(s):  
NATHAN BROWNLOWE ◽  
TOKE MEIER CARLSEN ◽  
MICHAEL F. WHITTAKER
Keyword(s):  
Directed Graphs ◽  
Orbit Equivalence ◽  
Graph Algebras ◽  
Orbit Equivalent ◽  
Markov Shifts ◽  
Graph Algebra ◽  
Arbitrary Graphs ◽  
Reverse Implication

We introduce the notion of orbit equivalence of directed graphs, following Matsumoto’s notion of continuous orbit equivalence for topological Markov shifts. We show that two graphs in which every cycle has an exit are orbit equivalent if and only if there is a diagonal-preserving isomorphism between their $C^{\ast }$-algebras. We show that it is necessary to assume that every cycle has an exit for the forward implication, but that the reverse implication holds for arbitrary graphs. As part of our analysis of arbitrary graphs $E$ we construct a groupoid ${\mathcal{G}}_{(C^{\ast }(E),{\mathcal{D}}(E))}$ from the graph algebra $C^{\ast }(E)$ and its diagonal subalgebra ${\mathcal{D}}(E)$ which generalises Renault’s Weyl groupoid construction applied to $(C^{\ast }(E),{\mathcal{D}}(E))$. We show that ${\mathcal{G}}_{(C^{\ast }(E),{\mathcal{D}}(E))}$ recovers the graph groupoid ${\mathcal{G}}_{E}$ without the assumption that every cycle in $E$ has an exit, which is required to apply Renault’s results to $(C^{\ast }(E),{\mathcal{D}}(E))$. We finish with applications of our results to out-splittings of graphs and to amplified graphs.

scholarly journals Graph varieties containing Murskii's groupoid

1989 ◽  
Vol 39 (2) ◽  
pp. 265-276
Author(s):  
R. Pöschel
Keyword(s):  
Finitely Based ◽  
Graph Algebras ◽  
Undirected Graphs ◽  
Graph Algebra ◽  
Subvariety Lattice ◽  
As Graph ◽  
Graph Varieties ◽  
Nonfinitely Based

In this paper varieties are investigated which are generated by graph algebras of undirected graphs and—in most cases—contain Murskii's groupoid (that is the graph algebra of the graph with two adjacent vertices and one loop). Though these varieties are inherently nonfinitely based, they can be finitely based as graph varieties (finitely graph based) like, for example, the varitey generated by Murskii's groupoid. Many examples of nonfinitely based graph varities containing Murskii's groupoid are given, too. Moreover, the coatoms in the subvariety lattice of the graph variety of all undirected graphs are described. There are two coatoms and they are finitely graph based.

scholarly journals -algebras of labelled graphs III—-theory computations

2015 ◽  
Vol 37 (2) ◽  
pp. 337-368 ◽  
Author(s):  
TERESA BATES ◽  
TOKE MEIER CARLSEN ◽  
DAVID PASK
Keyword(s):  
Uniqueness Theorem ◽  
Inductive Limit ◽  
Important Class ◽  
Graph Algebras ◽  
Gauge Invariant ◽  
Shift Spaces ◽  
Graph Algebra ◽  
Common Framework ◽  
Definition Of ◽  
Labelled Graphs

In this paper we give a formula for the$K$-theory of the$C^{\ast }$-algebra of a weakly left-resolving labelled space. This is done by realizing the$C^{\ast }$-algebra of a weakly left-resolving labelled space as the Cuntz–Pimsner algebra of a$C^{\ast }$-correspondence. As a corollary, we obtain a gauge-invariant uniqueness theorem for the$C^{\ast }$-algebra of any weakly left-resolving labelled space. In order to achieve this, we must modify the definition of the$C^{\ast }$-algebra of a weakly left-resolving labelled space. We also establish strong connections between the various classes of$C^{\ast }$-algebras that are associated with shift spaces and labelled graph algebras. Hence, by computing the$K$-theory of a labelled graph algebra, we are providing a common framework for computing the$K$-theory of graph algebras, ultragraph algebras, Exel–Laca algebras, Matsumoto algebras and the$C^{\ast }$-algebras of Carlsen. We provide an inductive limit approach for computing the$K$-groups of an important class of labelled graph algebras, and give examples.

IDENTITIES IN BIREGULAR LEFTMOST GRAPH VARIETIES OF TYPE (2,0)

2009 ◽  
Vol 02 (01) ◽  
pp. 1-17
Author(s):  
Apinant Anantpiniwatna ◽  
Tiang Poomsa-Ard
Keyword(s):  
Universal Algebra ◽  
Directed Graphs ◽  
Graph Algebras ◽  
Universal Algebras ◽  
Graph Algebra ◽  
Basic Concepts ◽  
Multiple Edges ◽  
Graph Varieties

Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type (2,0). We say that a graph G satisfies a term equation s ≈ t if the corresponding graph algebra A(G) satisfies s ≈ t. A class of graph algebras V is called a graph variety if V = ModgΣ where Σ is a subset of T(X) × T(X). A graph variety V' = ModgΣ' is called a biregular leftmost graph variety if Σ' is a set of biregular leftmost term equations. A term equation s ≈ t is called an identity in a variety V if G satisfies s ≈ t for all G ∈ V. In this paper we characterize identities in each biregular leftmost graph variety. For identities, varieties and other basic concepts of universal algebra see e.g. [1].

COMPUTATIONAL COMPLEXITY OF THE FINITE ALGEBRA MEMBERSHIP PROBLEM FOR VARIETIES

2002 ◽  
Vol 12 (06) ◽  
pp. 811-823 ◽  
Author(s):  
ZOLTÁN SZÉKELY
Keyword(s):  
Finite Algebra ◽  
Order Theory ◽  
Finite Basis ◽  
Membership Problem ◽  
Graph Algebras ◽  
Finite Basis Problem ◽  
Finite Algebras ◽  
First Order Theory ◽  
Open Questions ◽  
Graph Algebra

We exhibit finite algebras each generating a variety with NP-complete finite algebra membership problem. The smallest of these algebras is the flat graph algebra belonging to the tetrahedral graph, a graph of 6 vertices obtained by cutting and spreading out the surface of a tetrahedron on the plane. The sequence of graphs we use to build up our flat graph algebras is similar to the sequence exhibited by Wheeler in [36] , 1979, to describe the first order theory of k-colorable graphs. Graph algebras were introduced by Shallon in [34] , 1979, and investigated, among others, by Baker, McNulty and Werner in [2] , 1987. Flat algebras were constructed and used by McKenzie in [27] , 1996, to settle some open questions related to decidability, like Tarski's Finite Basis Problem. Flat graph algebras were also discussed by Willard in [37] , 1996, and Delić in [8] , 1998.

Graph variety generated by linear terms

2019 ◽  
Vol 12 (05) ◽  
pp. 1950074
Author(s):  
C. Manyuen ◽  
P. Jampachon ◽  
T. Poomsa-ard
Keyword(s):  
Directed Graphs ◽  
Linear Term ◽  
Graph Algebras ◽  
Universal Algebras ◽  
Type Formula ◽  
Graph Algebra ◽  
Multiple Edges ◽  
Graph Varieties

Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type [Formula: see text]. We say that a graph [Formula: see text] satisfies a term equation [Formula: see text] if the corresponding graph algebra [Formula: see text] satisfies [Formula: see text]. The set of all term equations [Formula: see text], which the graph [Formula: see text] satisfies, is denoted by [Formula: see text]. The class of all graph algebras satisfy all term equations in [Formula: see text] is called the graph variety generated by [Formula: see text] denoted by [Formula: see text]. A term is called a linear term if each variable which occurs in the term, occurs only once. A term equation [Formula: see text] is called a linear term equation if [Formula: see text] and [Formula: see text] are linear terms. This paper is devoted to a thorough investigation of graph varieties defined by linear term equations. In particular, we give a complete description of rooted graphs generating a graph variety described by linear term equations.

scholarly journals Associative spectra of graph algebras II

2021 ◽  
Author(s):  
Erkko Lehtonen ◽  
Tamás Waldhauser
Keyword(s):  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Graph Algebras ◽  
Arbitrary Graph ◽  
Graph Algebra ◽  
Necessary And Sufficient

AbstractA necessary and sufficient condition is presented for a graph algebra to satisfy a bracketing identity. The associative spectrum of an arbitrary graph algebra is shown to be either constant or exponentially growing.

ON ALGEBRA ISOMORPHISMS BETWEEN p-BANACH BEURLING ALGEBRAS

2021 ◽  
pp. 1-12
Author(s):  
PRAKASH A. DABHI ◽  
DARSHANA B. LIKHADA
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Discrete Groups ◽  
Discrete Semigroups ◽  
Beurling Algebras

Abstract Let $(G_1,\omega _1)$ and $(G_2,\omega _2)$ be weighted discrete groups and $0\lt p\leq 1$ . We characterise biseparating bicontinuous algebra isomorphisms on the p-Banach algebra $\ell ^p(G_1,\omega _1)$ . We also characterise bipositive and isometric algebra isomorphisms between the p-Banach algebras $\ell ^p(G_1,\omega _1)$ and $\ell ^p(G_2,\omega _2)$ and isometric algebra isomorphisms between $\ell ^p(S_1,\omega _1)$ and $\ell ^p(S_2,\omega _2)$ , where $(S_1,\omega _1)$ and $(S_2,\omega _2)$ are weighted discrete semigroups.

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