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.

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 Operator algebras with hyperarithmetic theory

2020 ◽  
Author(s):  
Isaac Goldbring ◽  
Bradd Hart
Keyword(s):  
Word Problem ◽  
Operator Algebras ◽  
Cuntz Algebra ◽  
Finitely Generated ◽  
C Algebra ◽  
Finitely Generated Group ◽  
C Algebras ◽  
Existentially Closed ◽  
Solvable Word Problem ◽  
Solvable Word

Abstract We show that the following operator algebras have hyperarithmetic theory: the hyperfinite II$_1$ factor $\mathcal R$, $L(\varGamma )$ for $\varGamma $ a finitely generated group with solvable word problem, $C^*(\varGamma )$ for $\varGamma $ a finitely presented group, $C^*_\lambda (\varGamma )$ for $\varGamma $ a finitely generated group with solvable word problem, $C(2^\omega )$ and $C(\mathbb P)$ (where $\mathbb P$ is the pseudoarc). We also show that the Cuntz algebra $\mathcal O_2$ has a hyperarithmetic theory provided that the Kirchberg embedding problems have affirmative answers. Finally, we prove that if there is an existentially closed (e.c.) II$_1$ factor (resp. $\textrm{C}^*$-algebra) that does not have hyperarithmetic theory, then there are continuum many theories of e.c. II$_1$ factors (resp. e.c. $\textrm{C}^*$-algebras).

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 Fiberwise KK-equivalence of continuous fields of C*-algebras

2008 ◽  
Vol 3 (2) ◽  
pp. 205-219 ◽  
Author(s):  
Marius Dadarlat
Keyword(s):  
Compact Space ◽  
Metrizable Space ◽  
Cuntz Algebra ◽  
Hilbert Cube ◽  
Locally Compact ◽  
Infinite Dimensional ◽  
C Algebras ◽  
Finite Dimensional ◽  
Compact Metrizable Space ◽  
Continuous Fields

AbstractLet A and B be separable nuclear continuous C(X)-algebras over a finite dimensional compact metrizable space X. It is shown that an element σ of the parametrized Kasparov group KKX(A,B) is invertible if and only all its fiberwise components σx ∈ KK(A(x),B(x)) are invertible. This criterion does not extend to infinite dimensional spaces since there exist nontrivial unital separable continuous fields over the Hilbert cube with all fibers isomorphic to the Cuntz algebra . Several applications to continuous fields of Kirchberg algebras are given. It is also shown that if each fiber of a separable nuclear continuous C(X)-algebra A over a finite dimensional locally compact space X satisfies the UCT, then A satisfies the UCT.

UNIVERSAL ALGEBRA OF SECTORS

2009 ◽  
Vol 19 (03) ◽  
pp. 347-371 ◽  
Author(s):  
KATSUNORI KAWAMURA
Keyword(s):  
Field Theory ◽  
Universal Algebra ◽  
Equivalence Classes ◽  
Cuntz Algebra ◽  
Quantum Field ◽  
Unitary Equivalence ◽  
C Algebra ◽  
Branching Laws ◽  
Common Unit ◽  
Algebraic Formulation

We show that a nontrivial example of universal algebra appears in quantum field theory. For a unital C *-algebra [Formula: see text], a sector is a unitary equivalence class of unital *-endomorphisms of [Formula: see text]. We show that the set [Formula: see text] of all sectors of [Formula: see text] is a universal algebra with an N-ary sum which is not reduced to any binary sum when [Formula: see text] includes the Cuntz algebra [Formula: see text] as a C *-subalgebra with common unit for N ≥ 3. Next we explain that the set [Formula: see text] of all unitary equivalence classes of unital *-representations of [Formula: see text] is a right module of [Formula: see text]. An essential algebraic formulation of branching laws of representations is given by using submodules of [Formula: see text]. As an application, we show that the action of [Formula: see text] on [Formula: see text] distinguishes elements of [Formula: see text].

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

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 Cartan subalgebras and the UCT problem, II

2020 ◽  
Vol 378 (1-2) ◽  
pp. 255-287
Author(s):  
Selçuk Barlak ◽  
Xin Li
Keyword(s):  
Algebraic Topology ◽  
Classification Theory ◽  
Cuntz Algebra ◽  
Crossed Products ◽  
C Algebra ◽  
C Algebras ◽  
Cyclic Groups ◽  
Open Questions ◽  
Cartan Subalgebras ◽  
Stably Finite

Abstract We study the connection between the UCT problem and Cartan subalgebras in C*-algebras. The UCT problem asks whether every separable nuclear C*-algebra satisfies the UCT, i.e., a noncommutative analogue of the classical universal coefficient theorem from algebraic topology. This UCT problem is one of the remaining major open questions in the structure and classification theory of simple nuclear C*-algebras. Since the class of separable nuclear C*-algebras is closed under crossed products by finite groups, it is a natural and important task to understand the behaviour of the UCT under such crossed products. We make a contribution towards a better understanding by showing that for certain approximately inner actions of finite cyclic groups on UCT Kirchberg algebras, the crossed products satisfy the UCT if and only if we can find Cartan subalgebras which are invariant under the actions of our finite cyclic groups. We also show that the class of actions we are able to treat is big enough to characterize the UCT problem, in the sense that every such action (even on a particular Kirchberg algebra, namely the Cuntz algebra $$\mathcal O_2$$ O 2 ) leads to a crossed product satisfying the UCT if and only if every separable nuclear C*-algebra satisfies the UCT. Our results rely on a new construction of Cartan subalgebras in certain inductive limit C*-algebras. This new tool turns out to be of independent interest. For instance, among other things, the second author has used it to construct Cartan subalgebras in all classifiable unital stably finite C*-algebras.

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].

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