Closure properties for the class of Cuntz-Krieger algebras Sara - PowerPoint PPT Presentation

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Faculty of Science Closure properties for the class of Cuntz-Krieger algebras Sara Arklint Department of Mathematical Sciences Canadian Operator Symposium, May 27-31, 2013 Slide 1/5

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Corners of Cuntz-Krieger algebras Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 2/5

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Corners of Cuntz-Krieger algebras Theorem (A-Ruiz) Let E be a countable directed graph. TFAE: Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 2/5

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Corners of Cuntz-Krieger algebras Theorem (A-Ruiz) Let E be a countable directed graph. TFAE: • C ∗ ( E ) is a Cuntz-Krieger algebra, Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 2/5

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Corners of Cuntz-Krieger algebras Theorem (A-Ruiz) Let E be a countable directed graph. TFAE: • C ∗ ( E ) is a Cuntz-Krieger algebra, • E is finite with no sinks, Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 2/5

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Corners of Cuntz-Krieger algebras Theorem (A-Ruiz) Let E be a countable directed graph. TFAE: • C ∗ ( E ) is a Cuntz-Krieger algebra, • E is finite with no sinks, • C ∗ ( E ) is unital and rank K 0 ( C ∗ ( E )) = rank K 1 ( C ∗ ( E )) . Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 2/5

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Corners of Cuntz-Krieger algebras Theorem (A-Ruiz) Let E be a countable directed graph. TFAE: • C ∗ ( E ) is a Cuntz-Krieger algebra, • E is finite with no sinks, • C ∗ ( E ) is unital and rank K 0 ( C ∗ ( E )) = rank K 1 ( C ∗ ( E )) . Theorem (A-Ruiz) Let A be a unital C ∗ -algebra and assume that A is stably isomorphic to a Cuntz-Krieger algebra. Then A is a Cuntz-Krieger algebra. Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 2/5

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Corners of Cuntz-Krieger algebras Theorem (A-Ruiz) Let E be a countable directed graph. TFAE: • C ∗ ( E ) is a Cuntz-Krieger algebra, • E is finite with no sinks, • C ∗ ( E ) is unital and rank K 0 ( C ∗ ( E )) = rank K 1 ( C ∗ ( E )) . Theorem (A-Ruiz) Let A be a unital C ∗ -algebra and assume that A is stably isomorphic to a Cuntz-Krieger algebra. Then A is a Cuntz-Krieger algebra. Corollary (A-Ruiz) Corners of Cuntz-Krieger algebras are Cuntz-Krieger algebras. Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 2/5

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Extensions of purely infinite Cuntz-Krieger algebras Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 3/5

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Extensions of purely infinite Cuntz-Krieger algebras Definition A C ∗ -algebra A looks like a purely infinite Cuntz-Krieger algebra if Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 3/5

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Extensions of purely infinite Cuntz-Krieger algebras Definition A C ∗ -algebra A looks like a purely infinite Cuntz-Krieger algebra if • A is unital, purely infinite, nuclear, separable, and of real rank zero, Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 3/5

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Extensions of purely infinite Cuntz-Krieger algebras Definition A C ∗ -algebra A looks like a purely infinite Cuntz-Krieger algebra if • A is unital, purely infinite, nuclear, separable, and of real rank zero, • A has finitely many ideals, Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 3/5

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Extensions of purely infinite Cuntz-Krieger algebras Definition A C ∗ -algebra A looks like a purely infinite Cuntz-Krieger algebra if • A is unital, purely infinite, nuclear, separable, and of real rank zero, • A has finitely many ideals, • for all I � J � A , the group K ∗ ( J / I ) is finitely generated, the group K 1 ( J / I ) is free, and rank K 0 ( J / I ) = rank K 1 ( J / I ) , Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 3/5

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Extensions of purely infinite Cuntz-Krieger algebras Definition A C ∗ -algebra A looks like a purely infinite Cuntz-Krieger algebra if • A is unital, purely infinite, nuclear, separable, and of real rank zero, • A has finitely many ideals, • for all I � J � A , the group K ∗ ( J / I ) is finitely generated, the group K 1 ( J / I ) is free, and rank K 0 ( J / I ) = rank K 1 ( J / I ) , • the simple subquotients of A are in the bootstrap class. Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 3/5

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Extensions of purely infinite Cuntz-Krieger algebras Definition A C ∗ -algebra A looks like a purely infinite Cuntz-Krieger algebra if • A is unital, purely infinite, nuclear, separable, and of real rank zero, • A has finitely many ideals, • for all I � J � A , the group K ∗ ( J / I ) is finitely generated, the group K 1 ( J / I ) is free, and rank K 0 ( J / I ) = rank K 1 ( J / I ) , • the simple subquotients of A are in the bootstrap class. Observation Consider a unital extension I ֒ → A ։ B . Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 3/5

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Extensions of purely infinite Cuntz-Krieger algebras Definition A C ∗ -algebra A looks like a purely infinite Cuntz-Krieger algebra if • A is unital, purely infinite, nuclear, separable, and of real rank zero, • A has finitely many ideals, • for all I � J � A , the group K ∗ ( J / I ) is finitely generated, the group K 1 ( J / I ) is free, and rank K 0 ( J / I ) = rank K 1 ( J / I ) , • the simple subquotients of A are in the bootstrap class. Observation Consider a unital extension I ֒ → A ։ B . If A is a purely infinite Cuntz-Krieger algebra, then Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 3/5

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Extensions of purely infinite Cuntz-Krieger algebras Definition A C ∗ -algebra A looks like a purely infinite Cuntz-Krieger algebra if • A is unital, purely infinite, nuclear, separable, and of real rank zero, • A has finitely many ideals, • for all I � J � A , the group K ∗ ( J / I ) is finitely generated, the group K 1 ( J / I ) is free, and rank K 0 ( J / I ) = rank K 1 ( J / I ) , • the simple subquotients of A are in the bootstrap class. Observation Consider a unital extension I ֒ → A ։ B . If A is a purely infinite Cuntz-Krieger algebra, then 1 B is a purely infinite Cuntz-Krieger algebra, Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 3/5

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f m a t h e m a t i c a l s c i e n c e s Extensions of purely infinite Cuntz-Krieger algebras Definition A C ∗ -algebra A looks like a purely infinite Cuntz-Krieger algebra if • A is unital, purely infinite, nuclear, separable, and of real rank zero, • A has finitely many ideals, • for all I � J � A , the group K ∗ ( J / I ) is finitely generated, the group K 1 ( J / I ) is free, and rank K 0 ( J / I ) = rank K 1 ( J / I ) , • the simple subquotients of A are in the bootstrap class. Observation Consider a unital extension I ֒ → A ։ B . If A is a purely infinite Cuntz-Krieger algebra, then 1 B is a purely infinite Cuntz-Krieger algebra, 2 I is stably isomorphic to a purely infinite Cuntz-Krieger algebra, Sara Arklint — Closure properties for the class of Cuntz-Krieger algebras — COSy, May 2013 Slide 3/5