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47
CuntzKrieger algebras of directed graphs
, 1996
"... We associate to each rowfinite directed graph E a universal CuntzKrieger C  algebra C (E), and study how the distribution of loops in E affects the structure of C (E). We prove that C (E) is AF if and only if E has no loops. We describe an exit condition (L) on loops in E which allow ..."
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Cited by 222 (48 self)
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We associate to each rowfinite directed graph E a universal CuntzKrieger C  algebra C (E), and study how the distribution of loops in E affects the structure of C (E). We prove that C (E) is AF if and only if E has no loops. We describe an exit condition (L) on loops in E which allows us to prove an analogue of the CuntzKrieger uniqueness theorem and give a characterisation of when C (E) is purely infinite. If the graph E satisfies (L) and is cofinal, then we have a dichotomy: if E has no loops, then C (E) is AF; if E has a loop, then C (E) is purely infinite.
The ideal structure of C ∗  algebras of infinite graphs
 Illinois J. Math
"... Abstract. We classify the gaugeinvariant ideals in the C ∗algebras of infinite directed graphs, and describe the quotients as graph algebras. We then use these results to identify the gaugeinvariant primitive ideals in terms of the structural properties of the graph, and describe the Ktheory of ..."
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Cited by 63 (7 self)
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Abstract. We classify the gaugeinvariant ideals in the C ∗algebras of infinite directed graphs, and describe the quotients as graph algebras. We then use these results to identify the gaugeinvariant primitive ideals in terms of the structural properties of the graph, and describe the Ktheory of the C ∗algebras of arbitrary infinite graphs. 1.
Graphs, groupoids and CuntzKrieger algebras
, 1996
"... We associate to each locally finite directed graph G two locally compact groupoids G and G(?). The unit space of G is the space of onesided infinite paths in G, and G(?) is the reduction of G to the space of paths emanating from a distinguished vertex ?. We show that under certain conditions the ..."
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Cited by 48 (18 self)
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We associate to each locally finite directed graph G two locally compact groupoids G and G(?). The unit space of G is the space of onesided infinite paths in G, and G(?) is the reduction of G to the space of paths emanating from a distinguished vertex ?. We show that under certain conditions their C algebras are Morita equivalent; the groupoid C algebra C (G) is the CuntzKrieger algebra of an infinite f0; 1g matrix defined by G, and that the algebras C (G(?)) contain the C algebras used by Doplicher and Roberts in their duality theory for compact groups. We then analyse the ideal structure of these groupoid C algebras using the general theory of Renault, and calculate their Ktheory. 1 Introduction Over the past fifteen years many C algebras and classes of C algebras have been given groupoid models. Here we consider locally finite directed graphs, which may have infinitely many vertices, but only finitely many edges in and out of each vertex. We associate ...
C*Algebras of Directed Graphs and Group Actions
, 1997
"... Given a free action of a group G on a directed graph E we show that the crossed product of C*(E), the universal C*algebra of E, by the induced action is strongly Morita equivalent to C*(E/G). Since every connected graph E may be expressed as the quotient of a tree T by an action of a free group G ..."
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Cited by 40 (11 self)
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Given a free action of a group G on a directed graph E we show that the crossed product of C*(E), the universal C*algebra of E, by the induced action is strongly Morita equivalent to C*(E/G). Since every connected graph E may be expressed as the quotient of a tree T by an action of a free group G we may use our results to show that C*(E) is strongly Morita equivalent to the crossed product C0 (@T ) \Theta G, where @T is a certain 0dimensional space canonically associated to the tree.
On the Ktheory of higher rank graph C*algebras
, 2008
"... Given a rowfinite kgraph Λ with no sources we investigate the Ktheory of the higher rank graph C ∗algebra, C ∗ (Λ). When k = 2 we are able to give explicit formulae to calculate the Kgroups of C ∗ (Λ). The Kgroups of C ∗ (Λ) for k> 2 can be calculated under certain circumstances and we cons ..."
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Cited by 32 (1 self)
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Given a rowfinite kgraph Λ with no sources we investigate the Ktheory of the higher rank graph C ∗algebra, C ∗ (Λ). When k = 2 we are able to give explicit formulae to calculate the Kgroups of C ∗ (Λ). The Kgroups of C ∗ (Λ) for k> 2 can be calculated under certain circumstances and we consider the case k =3. Weprovethat for arbitrary k, the torsionfree rank of K0(C ∗ (Λ)) and K1(C ∗ (Λ)) are equal when C ∗ (Λ) is unital, and for k = 2 we determine the position of
Continuous fields of C*algebras arising from extensions of tensor C*categories
"... The notion of extension of a given C*category C by a C*algebra A is introduced. In the commutative case A = C(Ω), the objects of the extension category are interpreted as fiber bundles over Ω of objects belonging ..."
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Cited by 16 (10 self)
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The notion of extension of a given C*category C by a C*algebra A is introduced. In the commutative case A = C(Ω), the objects of the extension category are interpreted as fiber bundles over Ω of objects belonging
C ∗ALGEBRAS OF LABELLED GRAPHS
, 2005
"... Abstract. We describe a class of C ∗algebras which simultaneously generalise the ultragraph algebras of Tomforde and the shift space C ∗algebras of Matsumoto. In doing so we shed some new light on the different C ∗algebras that may be associated to a shift space. Finally, we show how to associate ..."
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Cited by 16 (4 self)
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Abstract. We describe a class of C ∗algebras which simultaneously generalise the ultragraph algebras of Tomforde and the shift space C ∗algebras of Matsumoto. In doing so we shed some new light on the different C ∗algebras that may be associated to a shift space. Finally, we show how to associate a simple C ∗algebra to an irreducible sofic shift.
THE KTHEORY OF CUNTZKRIEGER ALGEBRAS FOR INFINITE MATRICES
, 1999
"... Abstract. We compute the Ktheory groups of the CuntzKrieger C ∗algebraOA associated to an infinite matrix A of zeros and ones. ..."
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Cited by 14 (0 self)
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Abstract. We compute the Ktheory groups of the CuntzKrieger C ∗algebraOA associated to an infinite matrix A of zeros and ones.
A.: A noncommutative AtiyahPatodiSinger index theorem
"... We investigate an extension of ideas of AtiyahPatodiSinger (APS) to a noncommutative geometry setting framed in terms of Kasparov modules. We use a mapping cone construction to relate odd index pairings to even index pairings with APS boundary conditions in the setting of KKtheory, generalising t ..."
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Cited by 11 (6 self)
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We investigate an extension of ideas of AtiyahPatodiSinger (APS) to a noncommutative geometry setting framed in terms of Kasparov modules. We use a mapping cone construction to relate odd index pairings to even index pairings with APS boundary conditions in the setting of KKtheory, generalising the commutative theory. We find that CuntzKreiger systems provide a natural class of examples for our construction and the index pairings coming from APS boundary conditions yield complete Ktheoretic information about certain graph C ∗algebras. Contents
On higher rank graph C ∗ algebras
, 2002
"... Abstract. Given a rowfinite kgraph Λ with no sources we investigate the Ktheory of the higher rank graph C ∗algebra, C ∗ (Λ). When k = 2 we are able to give explicit formulae to calculate the Kgroups of C ∗ (Λ). The Kgroups of C ∗ (Λ) for k> 2 can be calculated under certain circumstances. ..."
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Cited by 10 (1 self)
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Abstract. Given a rowfinite kgraph Λ with no sources we investigate the Ktheory of the higher rank graph C ∗algebra, C ∗ (Λ). When k = 2 we are able to give explicit formulae to calculate the Kgroups of C ∗ (Λ). The Kgroups of C ∗ (Λ) for k> 2 can be calculated under certain circumstances. We state that for all k, the torsionfree rank of K0(C ∗ (Λ)) and K1(C ∗ (Λ)) are equal when C ∗ (Λ) is unital, and we determine the position of the class of the unit of C ∗ (Λ) in K0(C ∗ (Λ)). 1.