Strongly Graded C*-algebras

<p>We investigate strongly graded C*-algebras. We focus on graph C*-algebras and explore the connection between graph C*-algebras and Leavitt path algebras, both of which are $\Z$-graded. It is known that a graphical condition called \emph{Condition (Y)} is necessary and sufficient for Leavitt path algebras to be strongly graded. In this thesis we prove this can be translated to the graph C*-algebra and prove that a graph C*-algebra associated to a row-finite graph is strongly graded if and only if Condition (Y) holds.</p>

A note on the regular ideals of Leavitt path algebras

10.1142/s0219498822502255 ◽

2021 ◽

pp. 2250225

Author(s):

Daniel Gonçalves ◽

Danilo Royer

Keyword(s):

Finite Graph ◽

Path Algebra ◽

Leavitt Path Algebra ◽

Arbitrary Graph ◽

C Algebras ◽

Path Algebras ◽

Regular Ideal ◽

Vertex Set ◽

Leavitt Path Algebras

We show that, for an arbitrary graph, a regular ideal of the associated Leavitt path algebra is also graded. As a consequence, for a row-finite graph, we obtain that the quotient of the associated Leavitt path by a regular ideal is again a Leavitt path algebra and that Condition (L) is preserved by quotients by regular ideals. Furthermore, we describe the vertex set of a regular ideal and make a comparison between the theory of regular ideals in Leavitt path algebras and in graph C*-algebras.

Free Product C* -algebras Associated with Graphs, Free Differentials, and Laws of Loops

Canadian Journal of Mathematics ◽

10.4153/cjm-2016-022-6 ◽

2017 ◽

Vol 69 (3) ◽

pp. 548-578 ◽

Author(s):

Michael Hartglass

Keyword(s):

Differential Calculus ◽

Von Neumann Algebras ◽

Sufficient Conditions ◽

Weighted Graph ◽

Positive Cone ◽

Von Neumann ◽

Planar Algebras ◽

C Algebra ◽

C Algebras ◽

AbstractWe study a canonical C* -algebra, 𝒮(Г,μ), that arises from a weighted graph (Г,μ), speci fic cases of which were previously studied in the context of planar algebras. We discuss necessary and sufficient conditions of the weighting that ensure simplicity and uniqueness of trace of 𝒮(Г,μ), and study the structure of its positive cone. We then study the *-algebra,𝒜, generated by the generators of 𝒮(Г,μ), and use a free differential calculus and techniques of Charlesworth and Shlyakhtenko as well as Mai, Speicher, and Weber to show that certain “loop” elements have no atoms in their spectral measure. After modifying techniques of Shlyakhtenko and Skoufranis to show that self adjoint elements x ∊ Mn(𝒜) have algebraic Cauchy transform, we explore some applications to eigenvalues of polynomials inWishart matrices and to diagrammatic elements in von Neumann algebras initially considered by Guionnet, Jones, and Shlyakhtenko.

Ideal-related $K$-theory for Leavitt path algebras and graph $C^*$-algebras

Indiana University Mathematics Journal ◽

10.1512/iumj.2013.62.5123 ◽

2013 ◽

Vol 62 (5) ◽

pp. 1587-1620 ◽

Cited By ~ 5

Author(s):

Efren Ruiz ◽

Mark Tomforde

Keyword(s):

C Algebras ◽

Path Algebras ◽

K Theory ◽

Leavitt Path Algebras

Leavitt path algebras satisfying a polynomial identity

10.1142/s0219498816500845 ◽

2016 ◽

Vol 15 (05) ◽

pp. 1650084 ◽

Author(s):

Jason P. Bell ◽

T. H. Lenagan ◽

Kulumani M. Rangaswamy

Keyword(s):

Polynomial Identity ◽

Finite Graph ◽

Infinite Graphs ◽

Leavitt Path Algebra ◽

Arbitrary Graph ◽

Graph Theoretic ◽

Path Algebras ◽

Leavitt Path Algebras ◽

Dimension One ◽

Kirillov Dimension

Leavitt path algebras [Formula: see text] of an arbitrary graph [Formula: see text] over a field [Formula: see text] satisfying a polynomial identity are completely characterized both in graph-theoretic and algebraic terms. When [Formula: see text] is a finite graph, [Formula: see text] satisfying a polynomial identity is shown to be equivalent to the Gelfand–Kirillov dimension of [Formula: see text] being at most one, though this is no longer true for infinite graphs. It is shown that, for an arbitrary graph [Formula: see text], the Leavitt path algebra [Formula: see text] has Gelfand–Kirillov dimension zero if and only if [Formula: see text] has no cycles. Likewise, [Formula: see text] has Gelfand–Kirillov dimension one if and only if [Formula: see text] contains at least one cycle, but no cycle in [Formula: see text] has an exit.

Discontinuous homomorphisms from C*-algebras

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100070195 ◽

1991 ◽

Vol 110 (1) ◽

pp. 147-150 ◽

Author(s):

D. W. B. Somerset

Keyword(s):

Banach Algebra ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

C Algebra ◽

C Algebras ◽

AbstractA necessary and sufficient condition is given for a unital C*-algebra A to admit a discontinuous homomorphism into a Banach algebra which is continuous on its centre. The condition is that A must have a Glimm ideal G such that the C*-algebra A/G admits a discontinuous homomorphism into a Banach algebra.

Full idempotents in Leavitt path algebras

10.1142/s0219498819500622 ◽

2019 ◽

Vol 18 (04) ◽

pp. 1950062

Author(s):

Ekrem Emre

Keyword(s):

Directed Graph ◽

Sufficient Conditions ◽

Path Algebra ◽

Necessary And Sufficient Conditions ◽

Leavitt Path Algebra ◽

Path Algebras ◽

Leavitt Path Algebras ◽

We give necessary and sufficient conditions on a directed graph [Formula: see text] for which the associated Leavit path algebra [Formula: see text] has at least one full idempotent. Also, we define [Formula: see text] sub-graphs of [Formula: see text] and show that [Formula: see text] has at least one full idempotent if and only if there is a sub-graph [Formula: see text] such that the associated Leavitt path algebra [Formula: see text] has at least one full idempotent.

On graded primitive leavitt path algebras

10.1142/s0219498821501735 ◽

2020 ◽

pp. 2150173

Author(s):

Kulumani M. Rangaswamy

Keyword(s):

Directed Graphs ◽

Sufficient Conditions ◽

Path Algebra ◽

Prime Rings ◽

Leavitt Path Algebra ◽

Von Neumann ◽

Path Algebras ◽

Von Neumann Regular ◽

Leavitt Path Algebras ◽

Graded primitive Leavitt path algebras of arbitrary directed graphs over a field [Formula: see text] are completely characterized by means of graphical conditions. Necessary and sufficient conditions are given under which a graded prime Leavitt path algebra becomes graded primitive and this leads to answering the graded version of a question of Kaplansky on von Neumann regular prime rings in the context of Leavitt path algebras.

Centers of Leavitt path algebras and their completions

10.1142/s0219498817500906 ◽

2017 ◽

Vol 16 (05) ◽

pp. 1750090 ◽

Cited By ~ 1

Author(s):

Adel Alahmadi ◽

Hamed Alsulami

Keyword(s):

Boolean Algebra ◽

Finite Graph ◽

Path Algebra ◽

Leavitt Path Algebra ◽

Underlying Graph ◽

Path Algebras ◽

Leavitt Path Algebras

In [9,10] Corrales Garcia, Barquero, Martin Gonzalez, Siles Molina, Solanilla Hernandez described the center of a Leavitt path algebra and characterized it in terms of the underlying graph. We offer a different characterization of the center. In particular, we prove that the Boolean algebra of central idempotents of a Leavitt path algebra of a finite graph is isomorphic to the Boolean algebra of finitary annihilator hereditary subsets of the graph.

Cohn–Leavitt path algebras and the invariant basis number property

10.1142/s0219498819500865 ◽

2019 ◽

Vol 18 (05) ◽

pp. 1950086 ◽

Author(s):

Müge Kanuni̇ ◽

Murad Özaydin

Keyword(s):

Finite Graph ◽

Path Algebra ◽

Necessary Condition ◽

Leavitt Path Algebra ◽

Morita Equivalent ◽

Path Algebras ◽

Invariant Basis ◽

K Theory ◽

Leavitt Path Algebras ◽

Invariant Basis Number

We give the necessary and sufficient condition for a separated Cohn–Leavitt path algebra of a finite digraph to have Invariant Basis Number (IBN). As a consequence, separated Cohn path algebras have IBN. We determine the non-stable K-theory of a corner ring in terms of the non-stable K-theory of the ambient ring. We give a necessary condition for a corner algebra of a separated Cohn–Leavitt path algebra of a finite graph to have IBN. We provide Morita equivalent rings which are non-IBN, but are of different types.