scholarly journals A topological isomorphism invariant for certain AF algebras

2005 ◽  
Vol 2005 (11) ◽  
pp. 1665-1673 ◽  
Author(s):  
Ryan J. Zerr
Keyword(s):  
Topological Space ◽  
Automorphism Groups ◽  
Bratteli Diagram ◽  
Topological Isomorphism ◽  
Dimension Group ◽  
Af Algebras

For certain AF algebras, a topological space is described which provides an isomorphism invariant for the algebras in this class. These AF algebras can be described in graphical terms by virtue of the existence of a certain type of Bratteli diagram, and the order-preserving automorphisms of the corresponding AF algebra's dimension group are then studied by utilizing this graph. This will also provide information about the automorphism groups of the corresponding AF algebras.

Bratteli diagrams via the De Concini–Procesi theorem

2020 ◽  
pp. 2050073
Author(s):  
Daniele Mundici
Keyword(s):  
Large Class ◽  
Word Problem ◽  
Bratteli Diagram ◽  
Von Neumann ◽  
Generators And Relations ◽  
The Core ◽  
Bratteli Diagrams ◽  
Group Presentations ◽  
Almost All ◽  
Af Algebras

An AF algebra [Formula: see text] is said to be an AF[Formula: see text] algebra if the Murray–von Neumann order of its projections is a lattice. Many, if not most, of the interesting classes of AF algebras existing in the literature are AF[Formula: see text] algebras. We construct an algorithm which, on input a finite presentation (by generators and relations) of the Elliott semigroup of an AF[Formula: see text] algebra [Formula: see text], generates a Bratteli diagram of [Formula: see text] We generalize this result to the case when [Formula: see text] has an infinite presentation with a decidable word problem, in the sense of the classical theory of recursive group presentations. Applications are given to a large class of AF algebras, including almost all AF algebras whose Bratteli diagram is explicitly described in the literature. The core of our main algorithms is a combinatorial-polyhedral version of the De Concini–Procesi theorem on the elimination of points of indeterminacy in toric varieties.

A Class of Compact Rigid 0-Dimensional Spaces

1969 ◽  
Vol 21 ◽  
pp. 817-821 ◽  
Author(s):  
F. W. Lozier
Keyword(s):  
Topological Space ◽  
Automorphism Groups ◽  
Affirmative Answer ◽  
Boolean Rings ◽  
Infinite Cardinality

A topological space is called “rigid” if its autohomeomorphism group is trivial. In (1), de Groot and McDowell showed that there are rigid, 0- dimensional spaces of arbitrarily high cardinality but left open the question of whether or not there are compact,rigid, 0-dimensional spaces of arbitrarily high cardinality, pointing out that an affirmative answer implies the existence of arbitrarily large Boolean rings with trivial automorphism groups. In this paper we construct a class of rigid, 0-dimensional spaces X αof arbitrary infinite cardinality and show that their Stone-Cech compactifications βX αare also rigid, thus answering the above question affirmatively.I would like to thank S. W. Willard, J. R. Isbell, and the referee for their careful readings of preliminary versions of this paper.

DIMENSION GROUPS, EMBEDDINGS AND PRESENTATIONS OF AF ALGEBRAS ASSOCIATED TO SOLVABLE LATTICE MODELS

1989 ◽  
Vol 04 (20) ◽  
pp. 1883-1890 ◽  
Author(s):  
DAVID E. EVANS ◽  
JEREMY D. GOULD
Keyword(s):  
Lattice Models ◽  
Path Algebra ◽  
Positive Cone ◽  
Point Of View ◽  
Conformal Field ◽  
Dimension Group ◽  
Dimension Groups ◽  
Ordered Ring ◽  
Af Algebras ◽  
The Ideal

If Γ is a graph, with distinguished vertex *, let A(Γ) denote the non-commutative path algebra on the space [Formula: see text] of semi-infinite paths in Γ beginning at *. Embeddings A(Γ1)→A(Γ2) of non-commutative AF algebras associated with graphs Γ1 and Γ2 are discussed from a dimension group point of view. For certain infinite T-shaped graphs, we have K0(A(Γ))≃ ℤ[t], with positive cone identified with {0}∪{P∈ℤ(t): P(λ)>0, λ∈(0,γ]}, where γ=γ(Γ)= ||Γ||−2<1/4. Hence for certain graphs there exists a unital homomorphism A(Γ1)→A(Γ2) if ||Γ1||=||Γ2||. For certain finite T-shaped graphs K0(A(Γ))≃ℤ[t]/<Q> where <Q> denotes the ideal generated by a polynomial Q=Q(Γ) which is essentially the characteristic polynomial of the graph Γ, and positive cone identified with {0}∪{f+<Q>: f(γ)>0} where γ=γ(Γ)=||Γ||−2. Hence there exists a unital homomorphism A(Γ1)→A(Γ2) if ||Γ1||=||Γ2||, and Q(Γ1) divides Q(Γ2). The structure of K0(A(Γ)) as an ordered ring is related to the fusion rules of rational conformal field theory. Moreover, for these T-shaped graphs there is an algebraic presentation which further illuminates the above embeddings. This presentation involves a new projection and a new relation in addition to those of Temperley-Lieb, and gives a rigidity above index four.

DIMENSION GROUPS AND EMBEDDINGS OF GRAPH ALGEBRAS

1994 ◽  
Vol 05 (03) ◽  
pp. 291-327 ◽  
Author(s):  
DAVID E. EVANS ◽  
JEREMY D. GOULD
Keyword(s):  
Path Algebra ◽  
Positive Cone ◽  
Point Of View ◽  
Graph Algebras ◽  
Conformal Field ◽  
Dimension Group ◽  
Dimension Groups ◽  
Ordered Ring ◽  
Af Algebras ◽  
The Ideal

If Γ is a graph, with distinguished vertex *, let A(Γ) denote the non-commutative path algebra on the space [Formula: see text] of semi-infinite paths in Γ beginning at *. We discuss embeddings A(Γ1) → A(Γ2) of AF algebras associated with graphs Γ1 and Γ2 from a dimension group point of view. For certain infinite T-shaped graphs, we have K0(A(Γ)) ≅ ℤ [t], with positive cone identified with {0}∪ {P ∈ ℤ [t]: P (λ) > 0, λ ∈ (0, γ]}, where γ = γ (Γ) =||Γ||−2 < 1/4. Hence for certain graphs there exists a unital homomorphism A(Γ1) → A(Γ2) if ||Γ1|| ≤ ||Γ2||. For certain finite T-shaped graphs K0 (A(Γ)) ≅ ℤ [t]/<Q> where <Q> denotes the ideal generated by a polynomial Q=Q(Γ) which is essentially the characteristic polynomial of the graph Γ and positive cone identified with {0}∪ {f + <Q>: f(γ) > 0} where γ = γ(Γ) = ||Γ||-2. Hence there exists a unital homomorphism A(Γ1) → A(Γ2) if ||Γ1|| = ||Γ2||, and Q(Γ2) divides Q(Γ1). The structure of K0(A(Γ)) as an ordered ring is related to the fusion rules of rational conformal field theory.

The Spatial Dimensions of Social Networks

2020 ◽  
pp. 367-383
Author(s):  
Zachary P. Neal
Keyword(s):  
Social Networks ◽  
Topological Space ◽  
Network Science ◽  
Spatial Dimensions

The first law of geography holds that everything is related to everything else, but near things are more related than distant things, where distance refers to topographical space. If a first law of network science exists, it would similarly hold that everything is related to everything else, but near things are more related than distant things, but where distance refers to topological space. Frequently these two laws collide, together holding that everything is related to everything else, but topographically and topologically near things are more related than topographically and topologically distant things. The focus of the spatial study of social networks lies in exploring a series of questions embedded in this combined law of geography and networks. This chapter explores the questions that have been asked and the answers that have been offered at the intersection of geography and networks.

Pentavalent 1-Transitive Digraphs with Non-Solvable Automorphism Groups

2020 ◽  
Vol 51 (4) ◽  
pp. 1919-1930
Author(s):  
Masoumeh Akbarizadeh ◽  
Mehdi Alaeiyan ◽  
Raffaele Scapellato
Keyword(s):  
Automorphism Groups
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