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33
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 ...
On the Morita equivalence of tensor algebras
 Proc. London Math. Soc
"... Our objective is two fold. First, we want to develop a notion of Morita equivalence for Ccorrespondences that guarantees that if two Ccorrespondences E and F are Morita equivalent, then the tensor algebras of E and F, T
E and T
F , are strongly Morita equivalent in the sense of [8], the Toepl ..."
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Cited by 46 (16 self)
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Our objective is two fold. First, we want to develop a notion of Morita equivalence for Ccorrespondences that guarantees that if two Ccorrespondences E and F are Morita equivalent, then the tensor algebras of E and F, T
E and T
F , are strongly Morita equivalent in the sense of [8], the Toeplitz algebras,
The equivariant Brauer group of a locally compact groupoid
, 1997
"... We define the Brauer group Br(G) of a locally compact groupoid G to be the set of Morita equivalence classes of pairs (A, α) consisting of an elementary C∗bundle A over G (0) satisfying Fell’s condition and an action α of G on A by ∗isomorphisms. When G is the transformation groupoid X × H, then ..."
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Cited by 39 (16 self)
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We define the Brauer group Br(G) of a locally compact groupoid G to be the set of Morita equivalence classes of pairs (A, α) consisting of an elementary C∗bundle A over G (0) satisfying Fell’s condition and an action α of G on A by ∗isomorphisms. When G is the transformation groupoid X × H, then Br(G) is the equivariant Brauer group BrH(X). In addition to proving that Br(G) is a group, we prove three isomorphism results. First we show that if G and H are equivalent groupoids, then Br(G) and Br(H) are isomorphic. This generalizes the result that if G and H are groups acting freely and properly on a space X, say G on the left and H on the right then BrG(X/H) and BrH(G\X) are isomorphic. Secondly we show that the subgroup Br0(G) of Br(G) consisting of classes [A, α] with A having trivial DixmierDouady invariant is isomorphic to a quotient E(G) of the collection Tw(G) of twists over G. Finally we prove that Br(G) is isomorphic to the inductive limit Ext(G, T) of the groups E(G X) where X varies over all principal G spaces X and G X is the imprimitivity groupoid associated to X.
The Shilov boundary of an operator space, and the characterization theorems
, 2000
"... We study operator spaces, operator algebras, and operator modules, from the point of view of the ‘noncommutative Shilov boundary’. In this attempt to utilize some ‘noncommutative Choquet theory’, we find that Hilbert C ∗modules and their properties, which we studied earlier in the operator space f ..."
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Cited by 34 (15 self)
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We study operator spaces, operator algebras, and operator modules, from the point of view of the ‘noncommutative Shilov boundary’. In this attempt to utilize some ‘noncommutative Choquet theory’, we find that Hilbert C ∗modules and their properties, which we studied earlier in the operator space framework, replace certain topological tools. We introduce certain multiplier operator algebras and C ∗algebras of an operator space, which generalize the algebras of adjointable operators on a C ∗module, and the ‘imprimitivity C ∗algebra’. It also generalizes a classical Banach space notion. This multiplier algebra plays a key role here. As applications of this perspective, we unify and strengthen several theorems characterizing operator algebras and modules. We also include some general notes on the ‘commutative case ’ of some of the topics we discuss, coming in part from joint work with Christian Le Merdy, about ‘function modules’.
Crossed products by dual coactions of groups and homogeneous spaces
 J. Operator Theory
"... Abstract. Mansfield showed how to induce representations of crossed products of C ∗algebras by coactions from crossed products by quotient groups and proved an imprimitivity theorem characterising these induced representations. We give an alternative construction of his bimodule in the case of dual ..."
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Cited by 29 (25 self)
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Abstract. Mansfield showed how to induce representations of crossed products of C ∗algebras by coactions from crossed products by quotient groups and proved an imprimitivity theorem characterising these induced representations. We give an alternative construction of his bimodule in the case of dual coactions, based on the symmetric imprimitivity theorem of the third author; this provides a more workable way of inducing representations of crossed products of C ∗algebras by dual coactions. The construction works for homogeneous spaces as well as quotient groups, and we prove an imprimitivity theorem for these induced representations. Coactions of groups on C∗algebras, and their crossed products, were introduced to make duality arguments available for the study of dynamical systems involving actions of nonabelian groups. For these to be effective, one needs to understand the representation theory of crossed products by coactions. The most powerful tool we have was provided by Mansfield [12]: he showed how to induce representations from crossed products by quotient groups, and proved an imprimitivity theorem which characterises these induced representations. Unfortunately, Mansfield’s construction is complicated and technical.
Imprimitivity for C ∗ coactions of nonamenable groups
 Math. Proc. Cambridge Philos. Soc. 123
, 1998
"... Abstract. We give a condition on a full coaction (A, G, δ) of a (possibly) nonamenable group G and a closed normal subgroup N of G which ensures that Mansfield imprimitivity works; i.e. that A × δ  G/N is Morita equivalent to A ×δ G × ˆ δ,r N. This condition obtains if N is amenable or δ is normal. ..."
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Cited by 15 (11 self)
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Abstract. We give a condition on a full coaction (A, G, δ) of a (possibly) nonamenable group G and a closed normal subgroup N of G which ensures that Mansfield imprimitivity works; i.e. that A × δ  G/N is Morita equivalent to A ×δ G × ˆ δ,r N. This condition obtains if N is amenable or δ is normal. It is preserved under Morita equivalence, inflation of coactions, the stabilization trick of Echterhoff and Raeburn, and on passing to twisted coactions. 1.
FREENESS OF ACTIONS OF FINITE GROUPS ON C*ALGEBRAS
, 2009
"... We describe some of the forms of freeness of group actions on noncommutative C*algebras that have been used, with emphasis on actions of finite groups. We give some indications of their strengths, weaknesses, applications, and relationships to each other. The properties discussed include the Rokh ..."
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Cited by 12 (3 self)
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We describe some of the forms of freeness of group actions on noncommutative C*algebras that have been used, with emphasis on actions of finite groups. We give some indications of their strengths, weaknesses, applications, and relationships to each other. The properties discussed include the Rokhlin property, Ktheoretic freeness, the tracial Rokhlin property, pointwise outerness, saturation, hereditary saturation, and the requirement that the strong Connes spectrum be the entire dual.
Imprimitivity For C*Coactions Of NonAmenable Groups
, 1996
"... . We give a condition on a full coaction (A; G; ffi ) of a (possibly) nonamenable group G and a closed normal subgroup N of G which ensures that Mansfield imprimitivity works; i.e. that A \Theta ffij G=N is Morita equivalent to A \Theta ffi G \Theta ffi ;r N . This condition obtains if N is amen ..."
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Cited by 12 (9 self)
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. We give a condition on a full coaction (A; G; ffi ) of a (possibly) nonamenable group G and a closed normal subgroup N of G which ensures that Mansfield imprimitivity works; i.e. that A \Theta ffij G=N is Morita equivalent to A \Theta ffi G \Theta ffi ;r N . This condition obtains if N is amenable or ffi is normal. It is preserved under Morita equivalence, inflation of coactions, the stabilization trick of Echterhoff and Raeburn, and on passing to twisted coactions. 1. Introduction One of Mackey's great gifts to mathematics is the imprimitivity theorem for locally compact groups, which Takesaki generalized in [Tak67] to C covariant systems. The modern (i.e., postRieffel) way to prove an imprimitivity theorem is to set up a (strong) Morita equivalence, and Green did this for Takesaki's theorem in [Gre78]. In the special case of abelian groups, Green's theorem can be viewed as saying that if ffi is an action of G on A, ffi is the dual action of G on the crossed product...
Strong Morita equivalence of higherdimensional noncommutative tori
, 2005
"... Abstract. We show that two C ∗algebraic noncommutative tori are strongly Morita equivalent if and only if they have isomorphic ordered K0groups and centers, extending N. C. Phillips’s result in the case that the algebras are simple. This is also generalized to the twisted group C ∗algebras of arb ..."
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Cited by 7 (2 self)
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Abstract. We show that two C ∗algebraic noncommutative tori are strongly Morita equivalent if and only if they have isomorphic ordered K0groups and centers, extending N. C. Phillips’s result in the case that the algebras are simple. This is also generalized to the twisted group C ∗algebras of arbitrary finitely generated abelian groups. 1.
Transformation group C*algebras with continuous trace
 II', J. Operator Theory
, 1984
"... We obtain several results characterizing when transformation group C*algebras have continuous trace. These results can be stated most succinctly when (G, L?) is second countable, and the stability groups are contained in a fixed abelian subgroup. In this case, C*(G, Q) has continuous trace if and o ..."
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Cited by 7 (1 self)
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We obtain several results characterizing when transformation group C*algebras have continuous trace. These results can be stated most succinctly when (G, L?) is second countable, and the stability groups are contained in a fixed abelian subgroup. In this case, C*(G, Q) has continuous trace if and only if the stability groups vary continuously on R and compact subsets of Q are wandering in an appropriate sense. In general, we must assume that the stability groups vary continuously, and if (G, J?) is not second countable, that the natural maps of G/S, onto G x are homeomorphisms for each x. Then C*(G, 0) has continuous trace if and only if compact subsets of 0 are wandering and an additionai C*algebra, constructed from the stability groups and 0, has continuous trace. 1.