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551
Localizing the Elliott conjecture at strongly selfabsorbing C∗algebras
, 2007
"... We formally introduce the concept of localizing the Elliott conjecture at a given strongly selfabsorbing C ∗algebra D; we also explain how the known classification theorems for nuclear C ∗algebras fit into this concept. As a new result in this direction, we employ recent results of Lin to show th ..."
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Cited by 46 (9 self)
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We formally introduce the concept of localizing the Elliott conjecture at a given strongly selfabsorbing C ∗algebra D; we also explain how the known classification theorems for nuclear C ∗algebras fit into this concept. As a new result in this direction, we employ recent results of Lin to show that (under a mild Ktheoretic condition) the class of separable, unital, simple C ∗algebras with locally finite decomposition rank and UCT, and for which projections separate traces, satisfies the Elliott conjecture localized at the Jiang–Su algebra Z. Our main result is formulated in a more general way; this allows us to outline a strategy to possibly remove the trace space condition as well as the Ktheory restriction entirely. When regarding both our result and the recent classification theorem of Elliott, Gong and Li as generalizations of the real rank zero case, the two approaches are perpendicular in a certain sense. The strategy to attack the general case aims at combining these two approaches. Our classification theorem covers simple ASH algebras for which projections
Metrics of Positive Scalar Curvature and Connections With Surgery
 Annals of Math. Studies
, 2001
"... this paper will be assumed to be smooth (C 1 ). For simplicity, we restrict attention to compact manifolds, although there are also plenty of interesting questions about complete metrics of positive scalar curvature on noncompact manifolds. At some points in the discussion, however, it will be ne ..."
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Cited by 45 (1 self)
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this paper will be assumed to be smooth (C 1 ). For simplicity, we restrict attention to compact manifolds, although there are also plenty of interesting questions about complete metrics of positive scalar curvature on noncompact manifolds. At some points in the discussion, however, it will be necessary to consider manifolds with boundary
Banach spaces with small spaces of operators
, 1997
"... Abstract. For a certain class of algebras A we give a method for constructing Banach spaces X such that every operator on X is close to an operator in A. This is used to produce spaces with a small amount of structure. We present several applications. Amongst them are constructions of a new prime Ba ..."
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Cited by 40 (2 self)
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Abstract. For a certain class of algebras A we give a method for constructing Banach spaces X such that every operator on X is close to an operator in A. This is used to produce spaces with a small amount of structure. We present several applications. Amongst them are constructions of a new prime Banach space, a space isomorphic to its subspaces of codimension two but not to its hyperplanes and a space isomorphic to its cube but not to its square. This paper is a continuation of [GM], in which a space X was constructed with the property that every operator from a subspace Y to X was of the form λi+S, where i is the inclusion map and S is strictly singular. Among the easy consequences of this fact are that X contains no unconditional basic sequence, and, more generally, that X is hereditarily
A KTheoretic Relative Index Theorem and CalliasType Dirac Operators
 Math. Ann
, 1995
"... We prove a relative index theorem for Dirac operators acting on sections of an A \Gamma C Clifford bundle, where A is some C algebra. We develop the index theory of complex and real Calliastype Dirac operators. ..."
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Cited by 40 (4 self)
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We prove a relative index theorem for Dirac operators acting on sections of an A \Gamma C Clifford bundle, where A is some C algebra. We develop the index theory of complex and real Calliastype Dirac operators.
Resolution of stringy singularities by noncommutative algebras
 JHEP 0106
"... Preprint typeset in JHEP style. PAPER VERSION ..."
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Dbranes on orbifolds with discrete torsion and topological obstruction
 JHEP 0005
, 2000
"... We find the orbifold analog of the topological relation recently found by Freed and Witten which restricts the allowed Dbrane configurations of Type II vacua with a topologically nontrivial flat Bfield. The result relies in Douglas proposal – which we derive from worldsheet consistency conditions ..."
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Cited by 38 (0 self)
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We find the orbifold analog of the topological relation recently found by Freed and Witten which restricts the allowed Dbrane configurations of Type II vacua with a topologically nontrivial flat Bfield. The result relies in Douglas proposal – which we derive from worldsheet consistency conditions – of embedding projective representations on open string ChanPaton factors when considering orbifolds with discrete torsion. The orbifold action on open strings gives a natural definition of the algebraic Ktheory group – using twisted cross products – responsible for measuring RamondRamond charges in orbifolds with discrete torsion. We show that the correspondence between fractional branes and RamondRamond fields follows in an interesting fashion from the way that discrete torsion is implemented on open and closed strings. January
Pullback and Pushout Constructions in C*algebra Theory
 JOURNAL OF FUNCTIONAL ANALYSIS 167, 243344 (1999)
, 1999
"... A systematic study of pullback and pushout diagrams is conducted in order to understand restricted direct sums and amalgamated free products of C*algebras. Particular emphasis is given to the relations with tensor products (both with the minimal and the maximal C*tensor norm). Thus it is shown tha ..."
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Cited by 36 (0 self)
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A systematic study of pullback and pushout diagrams is conducted in order to understand restricted direct sums and amalgamated free products of C*algebras. Particular emphasis is given to the relations with tensor products (both with the minimal and the maximal C*tensor norm). Thus it is shown that pullback and pushout diagrams are stable under tensoring with a fixed algebra and stable under crossed products with a fixed group. General tensor products between diagrams are also investigated. The relations between the theory of extensions and pullback and pushout diagrams are explored in some detail. The crowning result is that if three short exact sequences of C*algebras are given, with appropriate morphisms between the sequences allowing for pullback or pushout constructions at the levels of ideals, algebras and quotients, then the three new C*algebras will again form a short exact sequence under some mild extra conditions. As a generalization of a theorem of T. A. Loring it is shown that each morphism between a pair of C*algebras, combined with its extension to the stabilized algebras, gives rise to a pushout diagram. This result has applications to corona extendibility and conditional projectivity. Finally the pullback and pushout constructions are applied to the class of noncommutative CW complexes defined by (S. Eilers, T. A Loring, and
Computations of K and Ltheory of cocompact planar groups
 KTheory
"... Abstract. The verification of the isomorphism conjectures of Baum and Connes and Farrell and Jones for certain classes of groups is used to compute the algebraic KandLtheory and the topological Ktheory of cocompact planar groups ( = cocompact N.E.Cgroups) and of groups G appearing in an extensio ..."
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Cited by 34 (17 self)
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Abstract. The verification of the isomorphism conjectures of Baum and Connes and Farrell and Jones for certain classes of groups is used to compute the algebraic KandLtheory and the topological Ktheory of cocompact planar groups ( = cocompact N.E.Cgroups) and of groups G appearing in an extension 1 → Z n → G → π → 1whereπis a finite group and the conjugation πaction on Z n is free outside 0 ∈ Z n. These computations apply, for instance, to twodimensional crystallographic groups and cocompact Fuchsian groups.
The Maslov index, the spectral flow, and decomposition of manifolds
 Duke Math. J
, 1995
"... and a Clifford bundle M over M. The spectral flow of a smooth path of selfadjoint Dirac operators Dt: C(o) C(o) is the integer obtained by counting, with sign, the number of eigenvalues of D that cross 0 as varies; it is a homotopy invariant of the path (cf. [AS]). The aim of this paper is to descri ..."
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Cited by 34 (2 self)
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and a Clifford bundle M over M. The spectral flow of a smooth path of selfadjoint Dirac operators Dt: C(o) C(o) is the integer obtained by counting, with sign, the number of eigenvalues of D that cross 0 as varies; it is a homotopy invariant of the path (cf. [AS]). The aim of this paper is to describe the
The structure of crossed products of irrational rotation algebras by finite subgroups of SL2(Z)
, 2006
"... Let F ⊆ SL2(Z) be a finite subgroup (necessarily isomorphic to one of Z2, Z3, Z4, or Z6), and let F act on the irrational rotational algebra Aθ via the restriction of the canonical action of SL2(Z). Then the crossed product Aθ ⋊α F and the fixed point algebra AF θ are AF algebras. The same is true ..."
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Cited by 33 (12 self)
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Let F ⊆ SL2(Z) be a finite subgroup (necessarily isomorphic to one of Z2, Z3, Z4, or Z6), and let F act on the irrational rotational algebra Aθ via the restriction of the canonical action of SL2(Z). Then the crossed product Aθ ⋊α F and the fixed point algebra AF θ are AF algebras. The same is true for the crossed product and fixed point algebra of the flip action of Z2 on any simple ddimensional noncommutative torus AΘ. Along the way, we prove a number of general results which should have useful applications in other situations.