scholarly journals Standard Character Condition for Table Algebras

10.37236/385 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Amir Rahnamai Barghi ◽  
Javad Bagherian
Keyword(s):  
Association Scheme ◽  
C Algebra ◽  
C Algebras ◽  
Table Algebras ◽  
Adjacency Algebra ◽  
Regular Module ◽  
Standard Module ◽  
Table Algebra ◽  
Necessary And Sufficient ◽  
Standard Character

It is well known that the complex adjacency algebra $A$ of an association scheme has a specific module, namely the standard module, that contains the regular module of $A$ as a submodule. The character afforded by the standard module is called the standard character. In this paper we first define the concept of standard character for C-algebras and we say that a C-algebra has the standard character condition if it admits the standard character. Among other results we acquire a necessary and sufficient condition for a table algebra to originate from an association scheme. Finally, we prove that given a C-algebra admits the standard character and its all degrees are integers if and only if so its dual.

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

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 ◽  
Necessary And Sufficient

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.

Discontinuous homomorphisms from C*-algebras

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 ◽  
Necessary And Sufficient

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.

scholarly journals C*-ALGEBRAS THAT ARE ISOMORPHIC AFTER TENSORING AND FULL PROJECTIONS

2004 ◽  
Vol 47 (3) ◽  
pp. 659-668
Author(s):  
Kazunori Kodaka
Keyword(s):  
Matrix Algebra ◽  
Subject Classification ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Primary 46L05 ◽  
C Algebra ◽  
C Algebras ◽  
Necessary And Sufficient

AbstractLet $A$ be a unital $C^*$-algebra and for each $n\in\mathbb{N}$ let $M_n$ be the $n\times n$ matrix algebra over $\mathbb{C}$. In this paper we shall give a necessary and sufficient condition that there is a unital $C^*$-algebra $B$ satisfying $A\not\cong B$ but for which $A\otimes M_n\cong B\otimes M_n$ for some $n\in\mathbb{N}\setminus\{1\}$. Also, we shall give some examples of unital $C^*$-algebras satisfying the above property.AMS 2000 Mathematics subject classification: Primary 46L05

scholarly journals Strongly Graded C*-algebras

2021 ◽  
Author(s):  
◽  
Ellis Dawson
Keyword(s):  
Finite Graph ◽  
C Algebra ◽  
C Algebras ◽  
Path Algebras ◽  
Leavitt Path Algebras ◽  
Necessary And Sufficient

<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>

scholarly journals Strongly Graded C*-algebras

2021 ◽  
Author(s):  
◽  
Ellis Dawson
Keyword(s):  
Finite Graph ◽  
C Algebra ◽  
C Algebras ◽  
Path Algebras ◽  
Leavitt Path Algebras ◽  
Necessary And Sufficient

<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>

scholarly journals The Centre-Quotient Property and Weak Centrality for C*-Algebras

2020 ◽  
Author(s):  
Robert J Archbold ◽  
Ilja Gogić
Keyword(s):  
Central Element ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
C Algebra ◽  
C Algebras ◽  
Necessary And Sufficient ◽  
Equivalent Conditions ◽  
Quotient Property

Abstract We give a number of equivalent conditions (including weak centrality) for a general $C^*$-algebra to have the centre-quotient property. We show that every $C^*$-algebra $A$ has a largest weakly central ideal $J_{wc}(A)$. For an ideal $I$ of a unital $C^*$-algebra $A$, we find a necessary and sufficient condition for a central element of $A/I$ to lift to a central element of $A$. This leads to a characterisation of the set $V_A$ of elements of an arbitrary $C^*$-algebra $A$, which prevent $A$ from having the centre-quotient property. The complement $\textrm{CQ}(A):= A \setminus V_A$ always contains $Z(A)+J_{wc}(A)$ (where $Z(A)$ is the centre of $A$), with equality if and only if $A/J_{wc}(A)$ is abelian. Otherwise, $\textrm{CQ}(A)$ fails spectacularly to be a $C^*$-subalgebra of $A$.

scholarly journals Closable derivations of simple C*-algebras

1983 ◽  
Vol 24 (2) ◽  
pp. 181-183 ◽  
Author(s):  
Assadollah Niknam
Keyword(s):  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
C Algebra ◽  
C Algebras ◽  
Necessary And Sufficient

In this note we show that any derivation of a simple C*-algebra, whose range is not dense, is closable. We also derive a necessary and sufficient condition for a *-derivation of a C*-algebra, which is defined on the domain of a closed *- derivation, to be closed.

scholarly journals Inclusions of unital $C^*$-algebras of index-finite type with depth 2 induced by saturated actions of finite dimensional $C^*$-Hopf algebras

2009 ◽  
Vol 104 (2) ◽  
pp. 221
Author(s):  
Kazunori Kodaka ◽  
Yamotsu Teruya
Keyword(s):  
Hopf Algebra ◽  
Conditional Expectation ◽  
Hopf Algebras ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
Sigma H ◽  
Necessary And Sufficient

Let $B$ be a unital $C^*$-algebra and $H$ a finite dimensional $C^*$-Hopf algebra with its dual $C^*$-Hopf algebra $H^0$. We suppose that there is a saturated action of $H$ on $B$ and we denote by $A$ its fixed point $C^*$-subalgebra of $B$. Let $E$ be the canonical conditional expectation from $B$ onto $A$. In the present paper, we shall give a necessary and sufficient condition that there are a weak action of $H^0$ on $A$ and a unitary cocycle $\sigma$ of $H^0 \otimes H^0 $ to $A$ satisfying that there is an isomorphism $\pi$ of $A\rtimes_{\sigma}H^0 $ onto $B$, which is the twisted crossed product of $A$ by the weak action of $H^0$ on $A$ and the unitary cocycle $\sigma$, such that $F=E\circ \pi$, where $F$ is the canonical conditional expectation from $A\rtimes_{\sigma}H^0 $ onto $A$.

ON THE K-THEORY OF QUARTER-PLANE TOEPLITZ ALGEBRAS

1991 ◽  
Vol 02 (02) ◽  
pp. 195-204 ◽  
Author(s):  
EFTON PARK ◽  
CLAUDE SCHOCHET
Keyword(s):  
Spectral Sequence ◽  
Operator Theory ◽  
Sufficient Conditions ◽  
Theory Analysis ◽  
C Algebra ◽  
C Algebras ◽  
Quarter Plane ◽  
K Theory ◽  
Necessary And Sufficient

Given a C*-algebra A which is filtered by a collection of closed ideals Ai, there is a spectral sequence which relates the K-theory of A to the K-theory of the various quotient algebras Ai/Ai−1. The d1 differentials in this spectral sequence are familiar index invariants, but the higher differentials are not well-understood. Considering the case of Toeplitz C*-algebras associated with certain cones in Z2, it is shown that a d2 differential in the spectral sequence is non-trivial. This differential turns out to be an obstruction to a classical lifting problem in operator theory. Analysis of this obstruction leads to necessary and sufficient conditions for the lifting problem. It is hoped that this example will illuminate the role of higher differentials in the K-theory spectral sequence.

scholarly journals Limit $C^*$-algebras associated with an automorphism

2004 ◽  
Vol 95 (1) ◽  
pp. 101 ◽  
Author(s):  
Baruch Solel
Keyword(s):  
Fock Space ◽  
Direct Limit ◽  
Complete Classification ◽  
C Algebra ◽  
C Algebras ◽  
Product Presentation ◽  
Necessary And Sufficient ◽  
Deddens Algebras ◽  
Creation Operators

We present and study $C^*$-algebras generated by "periodic weighted creation operators" on the Fock space associated with an automorphism $\alpha$ on a $C^*$-algebra $A$. These algebras can be viewed as generalized Bunce-Deddens algebras associated with the automorphism and can be written as a certain direct limit. We prove a crossed product presentation for such an algebra and find a necessary and sufficient condition for it to be simple. In the case where the automorphism is induced by an irrational rotation (on C(T)) we compute the K-theory groups and obtain a complete classification of these algebras.

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