Abstract 2-groupoids C ∗ -algebras of 2-groupoids References C ∗ -algebras of 2-groupoids Massoud Amini Tarbiat Modares University Institute for Fundamental Researches (IPM) Banach Algebras 2013 Gothenburg, Sweden July 30, 2013 C ∗ -algebras of 2-groupoids Massoud Amini
Abstract 2-groupoids C ∗ -algebras of 2-groupoids References Table of contents 1 Abstract motivation 2 2-groupoids 2-categories algebraic 2-groupoids topological 2-groupoids and 2-Haar systems 3 C ∗ -algebras of 2-groupoids quasi-invariant measures full C ∗ -algebras induced representations and reduced C ∗ -algebras r -discrete principal 2-groupoids C ∗ -algebras of 2-groupoids Massoud Amini
Abstract 2-groupoids motivation C ∗ -algebras of 2-groupoids References Abstract We define topological 2-groupoids and study locally compact 2-groupoids with 2-Haar systems. We consider quasi-invariant measures on the sets of 1-arrows and unit space and build the corresponding vertical and horizontal modular functions. For a given 2-Haar system we construct the vertical and horizontal full C ∗ -algebras of a 2-groupoid and show that its is unique up to strong Morita equivalence, and make a correspondence between their bounded representations on Hilbert spaces and those of the 2-groupoid on Hilbert bundles. C ∗ -algebras of 2-groupoids Massoud Amini
Abstract 2-groupoids motivation C ∗ -algebras of 2-groupoids References Abstract We define topological 2-groupoids and study locally compact 2-groupoids with 2-Haar systems. We consider quasi-invariant measures on the sets of 1-arrows and unit space and build the corresponding vertical and horizontal modular functions. For a given 2-Haar system we construct the vertical and horizontal full C ∗ -algebras of a 2-groupoid and show that its is unique up to strong Morita equivalence, and make a correspondence between their bounded representations on Hilbert spaces and those of the 2-groupoid on Hilbert bundles. We show that representations of certain closed 2-subgroupoids are induced to representations of the 2-groupoid and use regular representation to build the vertical and horizontal reduced C ∗ -algebras of the 2-groupoid. We establish strong Morita equivalence between C ∗ -algebras of the 2-groupoid of composable pairs and those of the 1-arrows and unit space. We describe the reduced C ∗ -algebras of r-discrete principal 2-groupoids and find their ideals and masas. C ∗ -algebras of 2-groupoids Massoud Amini
Abstract 2-groupoids motivation C ∗ -algebras of 2-groupoids References Motivation In noncommutative geometry, certain quotient spaces are described by non-commutative C ∗ -algebras, when the symmetry groups of such quotient spaces are non Hausdorff, it is more appropriate to describe such symmetry groups and groupoids using crossed modules of groupoids (Buss-Meyer-Zhu, 2012). C ∗ -algebras of 2-groupoids Massoud Amini
Abstract 2-groupoids motivation C ∗ -algebras of 2-groupoids References Motivation In noncommutative geometry, certain quotient spaces are described by non-commutative C ∗ -algebras, when the symmetry groups of such quotient spaces are non Hausdorff, it is more appropriate to describe such symmetry groups and groupoids using crossed modules of groupoids (Buss-Meyer-Zhu, 2012). One motivating example is the gauge action on the irrational rotation algebra A ϑ , which is the universal C ∗ -algebra generated by two unitaries U and V satisfying the commutation relation UV = λ VU with λ := exp ( 2 π i ϑ ) . Since A ϑ is the crossed product C ( T ) ⋊ λ Z , for the canonical action of Z on T by n · z := λ n · z , it could be viewed as the noncommutatove analog of the non Hausdorff quotient space T /λ Z . This latter group acts on itself by translations, thus T /λ Z is a symmetry group of A ϑ . C ∗ -algebras of 2-groupoids Massoud Amini
Abstract 2-groupoids motivation C ∗ -algebras of 2-groupoids References Motivation More generally, one may define actions of crossed modules on C ∗ -algebras similar to the twisted actions in the sense of Philip Green (Green, 1978) and build crossed products for such actions. The resulting crossed product is functorial: If two actions are equivariantly Morita equivalent in a suitable sense, their crossed products are Morita–Rieffel equivalent C ∗ -algebras. C ∗ -algebras of 2-groupoids Massoud Amini
Abstract 2-groupoids motivation C ∗ -algebras of 2-groupoids References Motivation More generally, one may define actions of crossed modules on C ∗ -algebras similar to the twisted actions in the sense of Philip Green (Green, 1978) and build crossed products for such actions. The resulting crossed product is functorial: If two actions are equivariantly Morita equivalent in a suitable sense, their crossed products are Morita–Rieffel equivalent C ∗ -algebras. Crossed modules of discrete groups are used in homotopy theory to classify 2-connected spaces up to homotopy equivalence. They are equivalent to strict 2-groups (Baez, 1997, Noohi, 2007). C ∗ -algebras of 2-groupoids Massoud Amini
Abstract 2-groupoids motivation C ∗ -algebras of 2-groupoids References Motivation One could write every locally Hausdorff groupoid as the truncation of a Hausdorff topological weak 2-groupoid. Also the crossed modules of topological groupoids are equivalent to strict topological 2-groupoids. C ∗ -algebras of 2-groupoids Massoud Amini
Abstract 2-groupoids motivation C ∗ -algebras of 2-groupoids References Motivation One could write every locally Hausdorff groupoid as the truncation of a Hausdorff topological weak 2-groupoid. Also the crossed modules of topological groupoids are equivalent to strict topological 2-groupoids. For a Hausdorff étale groupoid G and the interior H ⊆ G of the set of loops (arrows with same source and target)in G , the quotient G / H is a locally Hausdorff, étale groupoid, and the pair ( G , H ) together with the embedding H → G and the conjugation action of G on H is a crossed module of topological groupoids. The corresponding C ∗ -algebra C ∗ ( G , H ) is the C ∗ -algebra of foliations in the sense of Alan Connes (Connes, 1982). The C ∗ -algebra of general (non Hausdorff) groupoids are studied in details by Jean Renault (Renault, 1980). C ∗ -algebras of 2-groupoids Massoud Amini
Abstract 2-categories 2-groupoids algebraic 2-groupoids C ∗ -algebras of 2-groupoids topological 2-groupoids and 2-Haar systems References strict 2-category We define a strict 2-category as a category enriched over categories. We adapt the notations and terminology of (Buss-Meyer-Zhu, 2013); see also (Baez, 1997). For two objects x and y of the first order category, we have a category of morphisms from x to y , and the composition of morphisms lifts to a bifunctor between these morphism categories. C ∗ -algebras of 2-groupoids Massoud Amini
� � � Abstract 2-categories 2-groupoids algebraic 2-groupoids C ∗ -algebras of 2-groupoids topological 2-groupoids and 2-Haar systems References strict 2-category We define a strict 2-category as a category enriched over categories. We adapt the notations and terminology of (Buss-Meyer-Zhu, 2013); see also (Baez, 1997). For two objects x and y of the first order category, we have a category of morphisms from x to y , and the composition of morphisms lifts to a bifunctor between these morphism categories. The arrows between objects u : x → y are called 1-morphisms. We write x = d ( u ) and y = r ( u ) . The arrows between arrows u y x , a v are called 2-morphisms (or bigons). We write u = d ( a ) , v = r ( a ) and x = d 2 ( a ) , y = r 2 ( a ) . C ∗ -algebras of 2-groupoids Massoud Amini
� � � � � � � � Abstract 2-categories 2-groupoids algebraic 2-groupoids C ∗ -algebras of 2-groupoids topological 2-groupoids and 2-Haar systems References Composition The category structure on the space of arrows x → y gives a vertical composition of 2-morphisms u u b �→ x . y x y v a · v b a w w C ∗ -algebras of 2-groupoids Massoud Amini
� � � � � � � � � � � � Abstract 2-categories 2-groupoids algebraic 2-groupoids C ∗ -algebras of 2-groupoids topological 2-groupoids and 2-Haar systems References Composition The vertical product a · v b is defined if r ( b ) = d ( a ) . The composition functor between the arrow categories gives a composition of 1-morphisms u v uv �→ x , z y x z which is defined if r ( v ) = d ( u ) , and a horizontal composition of 2-morphisms u 1 u 2 u 1 u 2 z y x �→ z x . a b a · h b v 1 v 2 v 1 v 2 The horizontal product a · h b is defined if r 2 ( b ) = d 2 ( a ) . C ∗ -algebras of 2-groupoids Massoud Amini
� � � � � � � � � � Abstract 2-categories 2-groupoids algebraic 2-groupoids C ∗ -algebras of 2-groupoids topological 2-groupoids and 2-Haar systems References Composition These three compositions are assumed to be associative and unital, with the same units for the vertical and horizontal products. The horizontal and vertical products commute: given a diagram u 1 u 2 a 1 a 2 z y x , v 1 v 2 b 1 b 2 w 1 w 2 composing first vertically and then horizontally or vice versa produces the same 2-morphism u 1 u 2 ⇒ v 1 v 2 . C ∗ -algebras of 2-groupoids Massoud Amini
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