scholarly journals Construction and pure infiniteness of $C^*$-algebras associated with lambda-graph systems

2005 ◽  
Vol 97 (1) ◽  
pp. 73 ◽  
Author(s):  
Kengo Matsumoto
Keyword(s):  
Hilbert Space ◽  
Bratteli Diagram ◽  
C Algebra ◽  
C Algebras ◽  
Shift Transformation ◽  
Labeled Graphs ◽  
Graph System ◽  
Creation Operators

A $\lambda$-graph system is a labeled Bratteli diagram with shift transformation. It is a generalization of finite labeled graphs and presents a subshift. In [16] the author has introduced a $C^*$-algebra $\mathcal{O}_{\mathfrak{L}}$ associated with a $\lambda$-graph system $\mathfrak{L}$ by using groupoid method as a generalization of the Cuntz-Krieger algebras. In this paper, we concretely construct the $C^*$-algebra $\mathcal{O}_{\mathfrak{L}}$ by using both creation operators and projections on a sub Fock Hilbert space associated with $\mathfrak{L}$. We also introduce a new irreducible condition on $\mathfrak{L}$ under which the $C^*$-algebra $\mathcal{O}_{\mathfrak{L}}$ becomes simple and purely infinite.

C*-algebras associated with presentations of subshifts ii. ideal structure and lambda-graph subsystems

2006 ◽  
Vol 81 (3) ◽  
pp. 369-385 ◽  
Author(s):  
Kengo Matsumoto
Keyword(s):  
Bratteli Diagram ◽  
Ideal Structure ◽  
View Point ◽  
Bijective Correspondence ◽  
C Algebras ◽  
Graph Theoretic ◽  
Labeled Graph ◽  
Shift Transformation ◽  
Labeled Graphs ◽  
Graph System

AbstractA λ-graph system is a labeled Bratteli diagram with shift transformation. It is a generalization of finite labeled graphs and presents a subshift. InDoc. Math.7 (2002) 1–30, the author constructed aC*-algebraO£associated with a λ-graph system £ from a graph theoretic view-point. If a λ-graph system comes from a finite labeled graph, the algebra becomes a Cuntz-Krieger algebra. In this paper, we prove that there is a bijective correspondence between the lattice of all saturated hereditary subsets of £ and the lattice of all ideals of the algebraO£, under a certain condition on £ called (II). As a result, the class of theC*-algebras associated with λ-graph systems under condition (II) is closed under quotients by its ideals.

FACTOR MAPS OF LAMBDA-GRAPH SYSTEMS AND INCLUSIONS OF C*-ALGEBRAS

2004 ◽  
Vol 15 (04) ◽  
pp. 313-339 ◽  
Author(s):  
KENGO MATSUMOTO
Keyword(s):  
Inductive Limit ◽  
C Algebra ◽  
C Algebras ◽  
Shift Transformation ◽  
Covering Graph ◽  
Labeled Graphs ◽  
Functorial Property ◽  
Graph Homomorphisms ◽  
Graph System ◽  
Factor Maps

A λ-graph system is a labeled Bratteli diagram with shift transformation. It is a generalization of finite labeled graphs and presents a subshift. In [Doc. Math. 7 (2002), 1–30], the author introduced a C*-algebra [Formula: see text] associated with a λ-graph system [Formula: see text] as a generalization of the Cuntz–Krieger algebras. In this paper, we study a functorial property between factor maps of λ-graph systems and inclusions of the associated C*-algebras with gauge actions. We prove that if there exists a surjective left-covering λ-graph system homomorphism [Formula: see text], there exists a unital embedding of the C*-algebra [Formula: see text] into the C*-algebra [Formula: see text] compatible to its gauge actions. We also show that a sequence of left-covering graph homomorphisms of finite labeled graphs gives rise to a λ-graph system such that the associated C*-algebra is an inductive limit of the Cuntz–Krieger algebras for the finite labeled graphs.

scholarly journals The irreducibility of C*-algebras acting on Hilbert C*-modules

Filomat ◽  
2016 ◽  
Vol 30 (9) ◽  
pp. 2425-2433
Author(s):  
Runliang Jiang
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Compact Operators ◽  
C Algebra ◽  
C Algebras ◽  
The Difference

Let B be a C*-algebra, E be a Hilbert B module and L(E) be the set of adjointable operators on E. Let A be a non-zero C*-subalgebra of L(E). In this paper, some new kinds of irreducibilities of A acting on E are introduced, which are all the generalizations of those associated to Hilbert spaces. The difference between these irreducibilities are illustrated by a number of counterexamples. It is concluded that for a full Hilbert B-module, these irreducibilities are all equivalent if and only if the underlying C*-algebra B is isomorphic to the C*-algebra of all compact operators on a Hilbert space.

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 On commutativity of C*-algebras

1987 ◽  
Vol 29 (1) ◽  
pp. 93-97 ◽  
Author(s):  
C.-S. Lin
Keyword(s):  
Hilbert Space ◽  
The Other ◽  
Nilpotent Operator ◽  
C Algebra ◽  
C Algebras ◽  
Adjoint Operators ◽  
Numerical Index

Two numerical characterizations of commutativity for C*-algebra (acting on the Hilbert space H) were given in [1]; one used the norms of self-adjoint operators in (Theorem 2), and the other the numerical index of (Theorem 3). In both cases the proofs were based on the result of Kaplansky which states that if the only nilpotent operator in is 0, then is commutative ([2] 2.12.21, p. 68). Of course the converse also holds.

DUALITY OF HOPF C*-ALGEBRAS

2002 ◽  
Vol 13 (09) ◽  
pp. 1009-1025 ◽  
Author(s):  
CHI-KEUNG NG
Keyword(s):  
Hilbert Space ◽  
Duality Theory ◽  
Fourier Algebra ◽  
C Algebra ◽  
C Algebras ◽  
Full Duality

In this paper, we study the duality theory of Hopf C*-algebras in a general "Hilbert-space-free" framework. Our particular interests are the "full duality" and the "reduced duality". In order to study the reduced duality, we define the interesting notion of Fourier algebra of a general Hopf C*-algebra. This study of reduced duality and Fourier algebra is found to be useful in the study of other aspects of Hopf C*-algebras (see e.g. [12–14]).

Stabilized $C^*$-algebras constructed from symbolic dynamical systems

2000 ◽  
Vol 20 (3) ◽  
pp. 821-841 ◽  
Author(s):  
KENGO MATSUMOTO
Keyword(s):  
Hilbert Space ◽  
Dynamical Systems ◽  
Tensor Product ◽  
Separable Hilbert Space ◽  
Dynamical Property ◽  
Compact Operators ◽  
Topological Conjugacy ◽  
Fock Spaces ◽  
C Algebras ◽  
Creation Operators

We construct stabilized $C^*$-algebras from subshifts by using the dynamical property of the symbolic dynamical systems. We prove that the construction is dynamical and the $C^*$-algebras are isomorphic to the tensor product $C^*$-algebras between the algebra of all compact operators on a separable Hilbert space and the $C^*$-algebras constructed from creation operators on sub-Fock spaces associated with the subshifts. We also prove that the gauge actions on the stabilized $C^*$-algebras are invariant for topological conjugacy as two-sided subshifts under some conditions. Hence, if two subshifts are topologically conjugate as two-sided subshifts, the associated stabilized $C^*$-algebras are isomorphic so that their K-groups are isomorphic.

C*-ALGEBRAS ASSOCIATED WITH SETS OF SEMIGROUPS OF ISOMETRIES

1991 ◽  
Vol 02 (03) ◽  
pp. 235-255 ◽  
Author(s):  
WILLIAM ARVESON
Keyword(s):  
Hilbert Space ◽  
Continuous Time ◽  
Separable Hilbert Space ◽  
Infinite Sequence ◽  
C Algebra ◽  
C Algebras

Let U0, U1, …, Un be a (finite or infinite) sequence of semigroups of isometries which act on the same separable Hilbert space H, n = 1, 2, …, ∞. {Uj} is said to be orthogonal if for all i ≠ j we have [Formula: see text] With every such sequence we associate a separable C*-algebra [Formula: see text]. These C*-algebras [Formula: see text], n = 1, 2, …, ∞, are the "continuous time" analogues of the Cuntz C*-algebras [Formula: see text], n = 2, 3, …, ∞, in the same sense that the Wiener-Hopf C*-algebra is the continuous time analogue of the Toeplitz C*-algebra. For example, we show that they are nuclear unitless C*-algebras which have no closed nontrivial ideals. Indeed, we show that each [Formula: see text] is stably isomorphic to one of the spectral C*-algebras which arise in the theory of E0-semigroups.

FRAMES IN HILBERT C*-MODULES AND MORITA EQUIVALENT C*-ALGEBRAS

2016 ◽  
Vol 59 (1) ◽  
pp. 1-10 ◽  
Author(s):  
MASSOUD AMINI ◽  
MOHAMMAD B. ASADI ◽  
GEORGE A. ELLIOTT ◽  
FATEMEH KHOSRAVI
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Discrete Spectrum ◽  
Compact Operators ◽  
Morita Equivalence ◽  
Hilbert Modules ◽  
C Algebra ◽  
C Algebras ◽  
Morita Equivalent ◽  
Continuous Trace

AbstractWe show that the property of a C*-algebra that all its Hilbert modules have a frame, in the case of σ-unital C*-algebras, is preserved under Rieffel–Morita equivalence. In particular, we show that a σ-unital continuous-trace C*-algebra with trivial Dixmier–Douady class, all of whose Hilbert modules admit a frame, has discrete spectrum. We also show this for the tensor product of any commutative C*-algebra with the C*-algebra of compact operators on any Hilbert space.

scholarly journals $C^*$-algebras arising from Dyck systems of topological Markov chains

2011 ◽  
Vol 109 (1) ◽  
pp. 31 ◽  
Author(s):  
Kengo Matsumoto
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Chain Rule ◽  
Irreducible Matrix ◽  
Partial Isometries ◽  
C Algebra ◽  
C Algebras ◽  
Graph System ◽  
Infinite Sequences

Let $A$ be an $N \times N$ irreducible matrix with entries in $\{0,1\}$. We define the topological Markov Dyck shift $D_A$ to be a nonsofic subshift consisting of bi-infinite sequences of the $2N$ brackets $(_1,\dots,(_N,)_1,\dots,)_N$ with both standard bracket rule and Markov chain rule coming from $A$. It is regarded as a subshift defined by the canonical generators $S_1^*,\dots, S_N^*, S_1,\dots, S_N$ of the Cuntz-Krieger algebra $\mathcal{O}_A$. We construct an irreducible $\lambda$-graph system $\mathcal{L}^{{\mathrm{Ch}}(D_A)}$ that presents the subshift $D_A$ so that we have an associated simple purely infinite $C^*$-algebra $\mathcal{O}_{\mathcal{L}^{{\mathrm{Ch}}(D_A)}}$. We prove that $\mathcal{O}_{\mathcal{L}^{{\mathrm{Ch}}(D_A)}}$ is a universal unique $C^*$-algebra subject to some operator relations among $2N$ generating partial isometries.

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