scholarly journals Schur Products

2014 ◽  
Author(s):  
Corneliu Constantinescu
Keyword(s):  
Clifford Algebras ◽  
Projective Representation ◽  
Schur Function ◽  
C Algebra ◽  
Infinite Dimensional ◽  
C Algebras ◽  
The Third ◽  
Normal Representation ◽  
Projective Representations ◽  
Factor Set

The projective representation of groups was introduced in 1904 by Issai Schur. It differs from the normal representation of groups by a twisting factor, which we call Schur function in this book and which is called sometimes normalized factor set in the literature (other names are also used). It starts with a discret group T and a Schur function f for T. This is a scalar valued function on T^2 satisfying the conditions f(1,1)=1 and |f(s,t)|=1, f(r,s)f(rs,t)=f(r,st)f(s,t) for all r,s,t in T. The projective representation of T twisted by f is a unital C*-subalgebra of the C*-algebra L(l^2(T)) of operators on the Hilbert space l^2(T). This reprezentation can be used in order to construct many examples of C*-algebras. By replacing the scalars R or C with an arbitrary unital (real or complex) C*-algebra E the field of applications is enhanced in an essential way. In this case l^2(T) is replaced by the Hilbert right E-module E tensor l^2(T) and L(l^2(T)) is replaced by the C*-algebra of adjointable operators on E tensor l^2(T). We call Schur product of E and T the resulting C*-algebra (in analogy to cross products which inspired the present construction). It opens a way to creat new K-theories (see the draft "Axiomatic K-theory for C*-algebras"). In a first chapter we introduce some results which are needed for this construction, which is developed in a second chapter. In the third chapter we present examples of C*-algebras obtained by this method. The classical Clifford Algebras (including the infinite dimensional ones) are C*-algbras which can be obtained by projective representations of certain groups. The last chapter of this book is dedicated to the generalization of these Clifford Algebras as an example of Schur products.

scholarly journals Large Irredundant Sets in Operator Algebras

2019 ◽  
Vol 72 (4) ◽  
pp. 988-1023
Author(s):  
Clayton Suguio Hida ◽  
Piotr Koszmider
Keyword(s):  
Von Neumann Algebras ◽  
Operator Algebras ◽  
Continuum Hypothesis ◽  
The Other ◽  
Open Problems ◽  
Von Neumann ◽  
C Algebra ◽  
Infinite Dimensional ◽  
C Algebras ◽  
The Continuum

AbstractA subset ${\mathcal{X}}$ of a C*-algebra ${\mathcal{A}}$ is called irredundant if no $A\in {\mathcal{X}}$ belongs to the C*-subalgebra of ${\mathcal{A}}$ generated by ${\mathcal{X}}\setminus \{A\}$. Separable C*-algebras cannot have uncountable irredundant sets and all members of many classes of nonseparable C*-algebras, e.g., infinite dimensional von Neumann algebras have irredundant sets of cardinality continuum.There exists a considerable literature showing that the question whether every AF commutative nonseparable C*-algebra has an uncountable irredundant set is sensitive to additional set-theoretic axioms, and we investigate here the noncommutative case.Assuming $\diamondsuit$ (an additional axiom stronger than the continuum hypothesis), we prove that there is an AF C*-subalgebra of ${\mathcal{B}}(\ell _{2})$ of density $2^{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D714}_{1}$ with no nonseparable commutative C*-subalgebra and with no uncountable irredundant set. On the other hand we also prove that it is consistent that every discrete collection of operators in ${\mathcal{B}}(\ell _{2})$ of cardinality continuum contains an irredundant subcollection of cardinality continuum.Other partial results and more open problems are presented.

scholarly journals $S$-regularity and the corona factorization property

2006 ◽  
Vol 99 (2) ◽  
pp. 204 ◽  
Author(s):  
D. Kucerovsky ◽  
P. W. Ng
Keyword(s):  
Fundamental Property ◽  
Sufficient Condition ◽  
Factorization Property ◽  
Stable Property ◽  
C Algebra ◽  
C Algebras ◽  
The Third ◽  
Explicit Reference ◽  
Stable Algebra ◽  
Extension Algebra

Stability is an important and fundamental property of $C^{*}$-algebras. Given a short exact sequence of $C^{*}$-algebras $0\longrightarrow B\longrightarrow E\longrightarrow A\longrightarrow 0$ where the ends are stable, the middle algebra may or may not be stable. We say that the first algebra, $B$, is $S$-regular if every extension of $B$ by a stable algebra $A$ has a stable extension algebra, $E$. Rördam has given a sufficient condition for $S$-regularity. We define a new condition, weaker than Rördam's, which we call the corona factorization property, and we show that the corona factorization property implies $S$-regularity. The corona factorization property originated in a study of the Kasparov $KK^1(A,B)$ group of extensions, however, we obtain our results without explicit reference to $KK$-theory. Our main result is that for a separable stable $C^{*}$-algebra $B$ the first two of the following properties (which we define later) are equivalent, and both imply the third. With additional hypotheses on the $C^{*}$-algebra, all three properties are equivalent. $B$ has the corona factorization property. Stability is a stable property for full hereditary subalgebras of $B$. $B$ is $S$-regular. We also show that extensions of separable stable $C^{*}$-algebras with the corona factorization property give extension algebras with the corona factorization property, extending the results of [9].

Factorization in the Invertible Group of a C*-Algebra

1997 ◽  
Vol 49 (6) ◽  
pp. 1188-1205 ◽  
Author(s):  
Michael J. Leen
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Compact Operators ◽  
Upper Bounds ◽  
C Algebra ◽  
Infinite Dimensional ◽  
Identity Component ◽  
C Algebras ◽  
Number Of Factors ◽  
Finite Products

AbstractIn this paper we consider the following problem: Given a unital C*- algebra A and a collection of elements S in the identity component of the invertible group of A, denoted inv0(A), characterize the group of finite products of elements of S. The particular C*-algebras studied in this paper are either unital purely infinite simple or of the form (A ⊗ K)+, where A is any C*-algebra and K is the compact operators on an infinite dimensional separable Hilbert space. The types of elements used in the factorizations are unipotents (1+ nilpotent), positive invertibles and symmetries (s2 = 1). First we determine the groups of finite products for each collection of elements in (A ⊗ K)+. Then we give upper bounds on the number of factors needed in these cases. The main result, which uses results for (A ⊗ K)+, is that for A unital purely infinite and simple, inv0(A) is generated by each of these collections of elements.

scholarly journals 2-LOCAL ISOMETRIES OF SOME OPERATOR ALGEBRAS

2002 ◽  
Vol 45 (2) ◽  
pp. 349-352 ◽  
Author(s):  
Lajos Molnár
Keyword(s):  
Operator Algebras ◽  
Linear Operators ◽  
Primary 47B49 ◽  
Similar Statement ◽  
Bounded Linear ◽  
C Algebra ◽  
Infinite Dimensional ◽  
C Algebras ◽  
Infinite Dimensional Hilbert Space ◽  
Local Isometry

AbstractAs a consequence of the main result of the paper we obtain that every 2-local isometry of the $C^*$-algebra $B(H)$ of all bounded linear operators on a separable infinite-dimensional Hilbert space $H$ is an isometry. We have a similar statement concerning the isometries of any extension of the algebra of all compact operators by a separable commutative $C^*$-algebra. Therefore, on those $C^*$-algebras the isometries are completely determined by their local actions on the two-point subsets of the underlying algebras.AMS 2000 Mathematics subject classification: Primary 47B49

Two-component symplectic universal characters and integrable hierarchies

2021 ◽  
pp. 2150045
Author(s):  
Chuanzhong Li
Keyword(s):  
Dynamic System ◽  
Clifford Algebras ◽  
Integrable Hierarchies ◽  
Schur Function ◽  
Tau Function ◽  
Free Fermions ◽  
Infinite Dimensional ◽  
Quadratic Equations ◽  
Function Solution ◽  
Two Component

In this paper, we first construct a symplectic Schur function solution to a newly defined two-component symplectic Kadomtsev–Petviashvili hierarchy. As a generalization of a two-component symplectic Schur function, we construct two-component symplectic universal characters which satisfy quadratic equations in an infinite-dimensional integrable dynamic system called a two-component symplectic universal character hierarchy. Then, we define a modified symplectic universal character hierarchy whose tau function can be represented by free fermions in Clifford algebras.

scholarly journals AUTOMATA COMPUTATION OF BRANCHING LAWS FOR ENDOMORPHISMS OF CUNTZ ALGEBRAS

2007 ◽  
Vol 17 (07) ◽  
pp. 1389-1409 ◽  
Author(s):  
KATSUNORI KAWAMURA
Keyword(s):  
Semigroup Theory ◽  
Automata Theory ◽  
De Bruijn Graph ◽  
Cuntz Algebra ◽  
C Algebra ◽  
Infinite Dimensional ◽  
C Algebras ◽  
Branching Laws ◽  
Branching Law ◽  
Mealy Machine

We study the representation theory of C*-algebras by using semigroup theory and automata theory. The Cuntz algebra [Formula: see text] is a finitely generated, infinite-dimensional, noncommutative C*-algebra. A certain class of cyclic representations of [Formula: see text] is characterized by words from the alphabet 1,…,N, which is called a cycle. A class of endomorphisms of [Formula: see text] is defined by polynomial functions in canonical generators and their conjugates. Such an endomorphism ρ of [Formula: see text] transforms a cycle π to π ◦ ρ which is a direct sum of cycles π1,…,πn unique up to unitary equivalence. The passage from π to π1,…,πn is called a branching law for ρ. In this article, we construct a Mealy machine from the endomorphism in order to compute its branching law. We show that the branching law is obtained as outputs from the machine for the input information of a given representation. Furthermore the actual computation of the branching law is executed by using a generalized de Bruijn graph associated with the Mealy machine.

THE STABLE AND THE REAL RANK OF ${\mathcal Z}$-ABSORBING C*-ALGEBRAS

2004 ◽  
Vol 15 (10) ◽  
pp. 1065-1084 ◽  
Author(s):  
MIKAEL RØRDAM
Keyword(s):  
Complex Numbers ◽  
Real Rank ◽  
Real Rank Zero ◽  
C Algebra ◽  
Infinite Dimensional ◽  
Matrix Algebras ◽  
C Algebras ◽  
Positive Elements ◽  
Stably Finite ◽  
Unital Simple

Suppose that A is a C*-algebra for which [Formula: see text], where [Formula: see text] is the Jiang–Su algebra: a unital, simple, stably finite, separable, nuclear, infinite-dimensional C*-algebra with the same Elliott invariant as the complex numbers. We show that: (i) The Cuntz semigroup W(A) of equivalence classes of positive elements in matrix algebras over A is almost unperforated. (ii) If A is exact, then A is purely infinite if and only if A is traceless. (iii) If A is separable and nuclear, then [Formula: see text] if and only if A is traceless. (iv) If A is simple and unital, then the stable rank of A is one if and only if A is finite. We also characterize when A is of real rank zero.

scholarly journals Nuclear subalgebras of UHF C*-algebras

1986 ◽  
Vol 29 (1) ◽  
pp. 97-100 ◽  
Author(s):  
R. J. Archbold ◽  
Alexander Kumjian
Keyword(s):  
Tensor Product ◽  
Inductive Limit ◽  
Inductive Limits ◽  
C Algebra ◽  
C Algebras ◽  
Finite Dimensional ◽  
The Family ◽  
Algebraic Tensor Product

A C*-algebra A is said to be approximately finite dimensional (AF) if it is the inductive limit of a sequence of finite dimensional C*-algebras(see [2], [5]). It is said to be nuclear if, for each C*-algebra B, there is a unique C*-norm on the *-algebraic tensor product A ⊗B [11]. Since finite dimensional C*-algebras are nuclear, and inductive limits of nuclear C*-algebras are nuclear [16];,every AF C*-algebra is nuclear. The family of nuclear C*-algebras is a large and well-behaved class (see [12]). The AF C*-algebras for a particularly tractable sub-class which has been completely classified in terms of the invariant K0 [7], [5].

scholarly journals Projective representations of quivers

2002 ◽  
Vol 31 (2) ◽  
pp. 97-101 ◽  
Author(s):  
Sangwon Park
Keyword(s):  
Projective Representation ◽  
Representations Of Quivers ◽  
Direct Sum ◽  
Projective Representations

We prove thatP1 →f P2is a projective representation of a quiverQ=•→•if and only ifP1andP2are projective leftR-modules,fis an injection, andf (P 1)⊂P 2is a summand. Then, we generalize the result so that a representationM1 →f1  M2  →f2⋯→fn−2  Mn−1→fn−1  Mnof a quiverQ=•→•→•⋯•→•→•is projective representation if and only if eachMiis a projective leftR-module and the representation is a direct sum of projective representations.

On C*-Algebras Associated with Subshifts

1997 ◽  
Vol 08 (03) ◽  
pp. 357-374 ◽  
Author(s):  
Kengo Matsumoto
Keyword(s):  
Symbolic Dynamics ◽  
Markov Shift ◽  
C Algebra ◽  
C Algebras ◽  
Markov Shifts ◽  
Universal Properties

We construct and study C*-algebras associated with subshifts in symbolic dynamics as a generalization of Cuntz–Krieger algebras for topological Markov shifts. We prove some universal properties for the C*-algebras and give a criterion for them to be simple and purely infinite. We also present an example of a C*-algebra coming from a subshift which is not conjugate to a Markov shift.

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