Introduction Negative results The unbounded picture Cuntz-Krieger algebras Finite summability in noncommutative geometry Magnus Go ff eng joint work with Bram Mesland Institut f¨ ur Analysis Leibniz Universit¨ at Hannover G¨ oteborg 2013-08-03 Banach Algebras and Applications Magnus Go ff eng joint work with Bram Mesland Finite summability in noncommutative geometry
Introduction Negative results The unbounded picture Cuntz-Krieger algebras Historical introduction 1 Atiyah-Singer’s index theorem (60’s) Magnus Go ff eng joint work with Bram Mesland Finite summability in noncommutative geometry
Introduction Negative results The unbounded picture Cuntz-Krieger algebras Historical introduction 1 Atiyah-Singer’s index theorem (60’s) 2 Abstract elliptic operators (Atiyah 1970) Magnus Go ff eng joint work with Bram Mesland Finite summability in noncommutative geometry
Introduction Negative results The unbounded picture Cuntz-Krieger algebras Historical introduction 1 Atiyah-Singer’s index theorem (60’s) 2 Abstract elliptic operators (Atiyah 1970) 3 Finite summability of Fredholm modules (Connes ∼ 1984) Magnus Go ff eng joint work with Bram Mesland Finite summability in noncommutative geometry
Introduction Negative results The unbounded picture Cuntz-Krieger algebras Historical introduction 1 Atiyah-Singer’s index theorem (60’s) 2 Abstract elliptic operators (Atiyah 1970) 3 Finite summability of Fredholm modules (Connes ∼ 1984) 4 Kasparov’s bivariant K -theory ( ∼ 1983) Magnus Go ff eng joint work with Bram Mesland Finite summability in noncommutative geometry
Introduction Negative results The unbounded picture Cuntz-Krieger algebras Historical introduction 1 Atiyah-Singer’s index theorem (60’s) 2 Abstract elliptic operators (Atiyah 1970) 3 Finite summability of Fredholm modules (Connes ∼ 1984) 4 Kasparov’s bivariant K -theory ( ∼ 1983) The finite summability problem 1 Given a C ∗ -algebra A , can any x ∈ K ∗ ( A ) be represented by a finitely summable Fredholm module? Magnus Go ff eng joint work with Bram Mesland Finite summability in noncommutative geometry
Introduction Negative results The unbounded picture Cuntz-Krieger algebras Historical introduction 1 Atiyah-Singer’s index theorem (60’s) 2 Abstract elliptic operators (Atiyah 1970) 3 Finite summability of Fredholm modules (Connes ∼ 1984) 4 Kasparov’s bivariant K -theory ( ∼ 1983) The finite summability problem 1 Given a C ∗ -algebra A , can any x ∈ K ∗ ( A ) be represented by a finitely summable Fredholm module? 2 If not, can one determine which x ∈ K ∗ ( A ) can be represented in this way? Magnus Go ff eng joint work with Bram Mesland Finite summability in noncommutative geometry
Introduction Negative results The unbounded picture Cuntz-Krieger algebras Some results in the negative direction Rave 2012 1 If x ∈ K 0 ( A ) can be represented by a K -cycle that is finitely summable on all of A , x can be represented by ( ⇡ , H , 0) where H is finite-dimensional. If x ∈ K 1 ( A ) can be represented by a K -cycle that is finitely summable on all of A , x = 0. 2 If Γ is a discrete group, A = C ∗ ( Γ ) and x ∈ K ∗ ( A ) can be represented by a K -cycle that is finitely summable on ` 1 ( Γ ), the same statement as above holds true. Magnus Go ff eng joint work with Bram Mesland Finite summability in noncommutative geometry
Introduction Negative results The unbounded picture Cuntz-Krieger algebras Some results in the negative direction Rave 2012 1 If x ∈ K 0 ( A ) can be represented by a K -cycle that is finitely summable on all of A , x can be represented by ( ⇡ , H , 0) where H is finite-dimensional. If x ∈ K 1 ( A ) can be represented by a K -cycle that is finitely summable on all of A , x = 0. 2 If Γ is a discrete group, A = C ∗ ( Γ ) and x ∈ K ∗ ( A ) can be represented by a K -cycle that is finitely summable on ` 1 ( Γ ), the same statement as above holds true. Puschnigg 2008 If Γ is a higher rank lattice (cofinite discrete subgroup of product of higher rank Lie group), any x ∈ K 0 ( C ∗ r ( Γ )) represented by a Fredholm module that is finitely summable on [ Γ ], satisfies x = 0. Magnus Go ff eng joint work with Bram Mesland Finite summability in noncommutative geometry
Introduction Negative results The unbounded picture Cuntz-Krieger algebras The nail in the co ffi n... A negative answer to the first question There is a x ∈ K 1 ( ⊕ ∞ n =1 C ( S 2 n − 1 )) \ { 0 } without a summable representative. Magnus Go ff eng joint work with Bram Mesland Finite summability in noncommutative geometry
Introduction Negative results The unbounded picture Cuntz-Krieger algebras The nail in the co ffi n... A negative answer to the first question There is a x ∈ K 1 ( ⊕ ∞ n =1 C ( S 2 n − 1 )) \ { 0 } without a summable representative. Proof. n =1 [ S 2 n − 1 ], where [ S 2 n − 1 ] ∈ K 1 ( C ( S 2 n − 1 )) denotes the Set x := P ∞ 1 fundamental class. Magnus Go ff eng joint work with Bram Mesland Finite summability in noncommutative geometry
Introduction Negative results The unbounded picture Cuntz-Krieger algebras The nail in the co ffi n... A negative answer to the first question There is a x ∈ K 1 ( ⊕ ∞ n =1 C ( S 2 n − 1 )) \ { 0 } without a summable representative. Proof. n =1 [ S 2 n − 1 ], where [ S 2 n − 1 ] ∈ K 1 ( C ( S 2 n − 1 )) denotes the Set x := P ∞ 1 fundamental class. A Theorem of Douglas-Voiculescu guarantees that the minimal summability 2 degree of x | C ( S 2 n − 1 ) is 2 n − 1. Hence x is not finitely summable on ⊕ alg n C ( S 2 n − 1 ). Magnus Go ff eng joint work with Bram Mesland Finite summability in noncommutative geometry
Introduction Negative results The unbounded picture Cuntz-Krieger algebras The nail in the co ffi n... A negative answer to the first question There is a x ∈ K 1 ( ⊕ ∞ n =1 C ( S 2 n − 1 )) \ { 0 } without a summable representative. Proof. n =1 [ S 2 n − 1 ], where [ S 2 n − 1 ] ∈ K 1 ( C ( S 2 n − 1 )) denotes the Set x := P ∞ 1 fundamental class. A Theorem of Douglas-Voiculescu guarantees that the minimal summability 2 degree of x | C ( S 2 n − 1 ) is 2 n − 1. Hence x is not finitely summable on ⊕ alg n C ( S 2 n − 1 ). Lemma n =1 C ( S 2 n − 1 ) is dense and holomorphically closed, ⊕ alg n C ( S 2 n − 1 ) ⊆ A . If A ⊆ ⊕ ∞ Magnus Go ff eng joint work with Bram Mesland Finite summability in noncommutative geometry
Introduction Negative results The unbounded picture Cuntz-Krieger algebras The nail in the co ffi n... A negative answer to the first question There is a x ∈ K 1 ( ⊕ ∞ n =1 C ( S 2 n − 1 )) \ { 0 } without a summable representative. Proof. n =1 [ S 2 n − 1 ], where [ S 2 n − 1 ] ∈ K 1 ( C ( S 2 n − 1 )) denotes the Set x := P ∞ 1 fundamental class. A Theorem of Douglas-Voiculescu guarantees that the minimal summability 2 degree of x | C ( S 2 n − 1 ) is 2 n − 1. Hence x is not finitely summable on ⊕ alg n C ( S 2 n − 1 ). Lemma n =1 C ( S 2 n − 1 ) is dense and holomorphically closed, ⊕ alg n C ( S 2 n − 1 ) ⊆ A . If A ⊆ ⊕ ∞ Magnus Go ff eng joint work with Bram Mesland Finite summability in noncommutative geometry
Introduction Negative results The unbounded picture Cuntz-Krieger algebras Spectral triples Spectral triples A spectral triple is a triple ( A , H , D ) where H = H + ⊕ H − is a graded Hilbert space, A ⊆ B ( H ) ev is a ∗ -subalgebra and D is a self-adjoint odd operator such that ( i + D ) − 1 ∈ K ( H ) Magnus Go ff eng joint work with Bram Mesland Finite summability in noncommutative geometry
Introduction Negative results The unbounded picture Cuntz-Krieger algebras Spectral triples Spectral triples A spectral triple is a triple ( A , H , D ) where H = H + ⊕ H − is a graded Hilbert space, A ⊆ B ( H ) ev is a ∗ -subalgebra and D is a self-adjoint odd operator such that ( i + D ) − 1 ∈ K ( H ) and ∀ a ∈ A : a Dom ( D ) ⊆ Dom ( D ) and [ D , a ] ∈ B ( H ) . Magnus Go ff eng joint work with Bram Mesland Finite summability in noncommutative geometry
Introduction Negative results The unbounded picture Cuntz-Krieger algebras Spectral triples Spectral triples A spectral triple is a triple ( A , H , D ) where H = H + ⊕ H − is a graded Hilbert space, A ⊆ B ( H ) ev is a ∗ -subalgebra and D is a self-adjoint odd operator such that ( i + D ) − 1 ∈ K ( H ) and ∀ a ∈ A : a Dom ( D ) ⊆ Dom ( D ) and [ D , a ] ∈ B ( H ) . If ( i + D ) − 1 ∈ L p ( H ), ( A , H , D ) is said to be p -summable. Magnus Go ff eng joint work with Bram Mesland Finite summability in noncommutative geometry
Introduction Negative results The unbounded picture Cuntz-Krieger algebras Spectral triples Spectral triples A spectral triple is a triple ( A , H , D ) where H = H + ⊕ H − is a graded Hilbert space, A ⊆ B ( H ) ev is a ∗ -subalgebra and D is a self-adjoint odd operator such that ( i + D ) − 1 ∈ K ( H ) and ∀ a ∈ A : a Dom ( D ) ⊆ Dom ( D ) and [ D , a ] ∈ B ( H ) . If ( i + D ) − 1 ∈ L p ( H ), ( A , H , D ) is said to be p -summable. The spectral triples refine analytic K -cycles from a conformal noncommutative geometry, to a metric noncommutative geometry. Magnus Go ff eng joint work with Bram Mesland Finite summability in noncommutative geometry
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