What is Noncommutative Geometry ? How a geometry can be commutative and why mine is not Alessandro Rubin Junior Math Days 2019/20 SISSA - Scuola Internazionale Superiore di Studi Avanzati This means that the algebra C∞(M) contains enough information to codify the whole geometry of the manifold: 1. Vector fields: linear derivations of C∞(M) 2. Differential 1-forms: C∞(M)-linear forms on vector fields 3. ... Question Do we really need a manifold’s points to study it ? Do we really use the commutativity of the algebra C∞(M) to define the aforementioned objects ? Doing Geometry Without a Geometric Space Theorem Two smooth manifolds M, N are diffeomorphic if and only if their algebras of smooth functions C∞(M) and C∞(N) are isomorphic. 1/41 Question Do we really need a manifold’s points to study it ? Do we really use the commutativity of the algebra C∞(M) to define the aforementioned objects ? Doing Geometry Without a Geometric Space Theorem Two smooth manifolds M, N are diffeomorphic if and only if their algebras of smooth functions C∞(M) and C∞(N) are isomorphic. This means that the algebra C∞(M) contains enough information to codify the whole geometry of the manifold: 1. Vector fields: linear derivations of C∞(M) 2. Differential 1-forms: C∞(M)-linear forms on vector fields 3. ... 1/41 Do we really use the commutativity of the algebra C∞(M) to define the aforementioned objects ? Doing Geometry Without a Geometric Space Theorem Two smooth manifolds M, N are diffeomorphic if and only if their algebras of smooth functions C∞(M) and C∞(N) are isomorphic. This means that the algebra C∞(M) contains enough information to codify the whole geometry of the manifold: 1. Vector fields: linear derivations of C∞(M) 2. Differential 1-forms: C∞(M)-linear forms on vector fields 3. ... Question Do we really need a manifold’s points to study it ? 1/41 Doing Geometry Without a Geometric Space Theorem Two smooth manifolds M, N are diffeomorphic if and only if their algebras of smooth functions C∞(M) and C∞(N) are isomorphic. This means that the algebra C∞(M) contains enough information to codify the whole geometry of the manifold: 1. Vector fields: linear derivations of C∞(M) 2. Differential 1-forms: C∞(M)-linear forms on vector fields 3. ... Question Do we really need a manifold’s points to study it ? Do we really use the commutativity of the algebra C∞(M) to define the aforementioned objects ? 1/41 They are all anagrams of the letters: adegi2l2no2ruvy Moral of the Story Commutativity ⇒ Loss of Information Why Consider Noncommutativity? Consider the following sentences: Darling I love you Leaving your idol Avoiding our yell 2/41 Moral of the Story Commutativity ⇒ Loss of Information Why Consider Noncommutativity? Consider the following sentences: Darling I love you Leaving your idol Avoiding our yell They are all anagrams of the letters: adegi2l2no2ruvy 2/41 Why Consider Noncommutativity? Consider the following sentences: Darling I love you Leaving your idol Avoiding our yell They are all anagrams of the letters: adegi2l2no2ruvy Moral of the Story Commutativity ⇒ Loss of Information 2/41 If A is commutative we want back all the known machinery. Noncommutative manifolds do NOT exist. The previous method can be adapted to study also: A topological space X and the algebra C(X) A measure space (X, µ) and the algebra L∞(X, µ) ... Commutative Geometries Fundamental Idea of Noncommutative Geometry Replace C∞(M) with a possibly noncommutative algebra A and regard A as the algebra of functions on a "noncommutative smooth manifold". M C∞(M) Vector Fields, Diff. Forms, ... ? A Analogous Algebraic Constructions 3/41 Noncommutative manifolds do NOT exist. The previous method can be adapted to study also: A topological space X and the algebra C(X) A measure space (X, µ) and the algebra L∞(X, µ) ... Commutative Geometries Fundamental Idea of Noncommutative Geometry Replace C∞(M) with a possibly noncommutative algebra A and regard A as the algebra of functions on a "noncommutative smooth manifold". M C∞(M) Vector Fields, Diff. Forms, ... ? A Analogous Algebraic Constructions If A is commutative we want back all the known machinery. 3/41 The previous method can be adapted to study also: A topological space X and the algebra C(X) A measure space (X, µ) and the algebra L∞(X, µ) ... Commutative Geometries Fundamental Idea of Noncommutative Geometry Replace C∞(M) with a possibly noncommutative algebra A and regard A as the algebra of functions on a "noncommutative smooth manifold". M C∞(M) Vector Fields, Diff. Forms, ... ? A Analogous Algebraic Constructions If A is commutative we want back all the known machinery. Noncommutative manifolds do NOT exist. 3/41 Commutative Geometries Fundamental Idea of Noncommutative Geometry Replace C∞(M) with a possibly noncommutative algebra A and regard A as the algebra of functions on a "noncommutative smooth manifold". M C∞(M) Vector Fields, Diff. Forms, ... ? A Analogous Algebraic Constructions If A is commutative we want back all the known machinery. Noncommutative manifolds do NOT exist. The previous method can be adapted to study also: A topological space X and the algebra C(X) A measure space (X, µ) and the algebra L∞(X, µ) ... 3/41 Table of Contents 1. Noncommutative Topology 2. Noncommutative Measure Theory 3. Noncommutative Manifolds 4. The Noncommutative Torus 5. Some Applications to Physics 4/41 Noncommutative Topology Example Let X be a topological space. The set Cb(X) of continuous and ∗ bounded functions f : X → C is a C -algebra where the product is given pointwise, the norm and the involution are defined by kf k = sup |f (x)| (f ∗)(x) = f (x) x∈X C ∗-Algebras and Functions Definition AC ∗-algebra is a complex algebra A endowed with a norm A and an anti-linear map ∗ : A → A such that they satisfy some technical axioms and the following relation: ka∗ak = kak2 ∀a ∈ A 5/41 C ∗-Algebras and Functions Definition AC ∗-algebra is a complex algebra A endowed with a norm A and an anti-linear map ∗ : A → A such that they satisfy some technical axioms and the following relation: ka∗ak = kak2 ∀a ∈ A Example Let X be a topological space. The set Cb(X) of continuous and ∗ bounded functions f : X → C is a C -algebra where the product is given pointwise, the norm and the involution are defined by kf k = sup |f (x)| (f ∗)(x) = f (x) x∈X 5/41 Theorem (First Gelfand-Najmark Theorem) Let X be a locally compact Hausdorff topological space. The ∗ space C0(X) is a commutative C -algebra. Let A be a commutative C ∗-algebra. There exists a locally compact Hausdorff topological space X such that A 'C0(X). The Gelfand-Najmark Correspondence Definition Let X be a locally compact Hausdorff topological space. The algebra C0(X) of functions that vanish at infinity is the norm closure of the functions with compact support Cc (X). 6/41 Let A be a commutative C ∗-algebra. There exists a locally compact Hausdorff topological space X such that A 'C0(X). The Gelfand-Najmark Correspondence Definition Let X be a locally compact Hausdorff topological space. The algebra C0(X) of functions that vanish at infinity is the norm closure of the functions with compact support Cc (X). Theorem (First Gelfand-Najmark Theorem) Let X be a locally compact Hausdorff topological space. The ∗ space C0(X) is a commutative C -algebra. 6/41 The Gelfand-Najmark Correspondence Definition Let X be a locally compact Hausdorff topological space. The algebra C0(X) of functions that vanish at infinity is the norm closure of the functions with compact support Cc (X). Theorem (First Gelfand-Najmark Theorem) Let X be a locally compact Hausdorff topological space. The ∗ space C0(X) is a commutative C -algebra. Let A be a commutative C ∗-algebra. There exists a locally compact Hausdorff topological space X such that A 'C0(X). 6/41 There is a bijective correspondence between maximal modular ideals I ⊆ A and the quotient maps γ : A → A/I ' C. Under this identification, X is endowed with the weak-∗ topology. The isomorphism is given by the Gelfand Transform Γ: A −→ C0(X) a 7→ ba where ba : X → C is the point evaluation ba(γ) = γ(a). How is This Possible ? The correspondence of the previous theorem is very well known: Points of X are given by the maximal modular ideals of A. 7/41 The isomorphism is given by the Gelfand Transform Γ: A −→ C0(X) a 7→ ba where ba : X → C is the point evaluation ba(γ) = γ(a). How is This Possible ? The correspondence of the previous theorem is very well known: Points of X are given by the maximal modular ideals of A. There is a bijective correspondence between maximal modular ideals I ⊆ A and the quotient maps γ : A → A/I ' C. Under this identification, X is endowed with the weak-∗ topology. 7/41 How is This Possible ? The correspondence of the previous theorem is very well known: Points of X are given by the maximal modular ideals of A. There is a bijective correspondence between maximal modular ideals I ⊆ A and the quotient maps γ : A → A/I ' C. Under this identification, X is endowed with the weak-∗ topology. The isomorphism is given by the Gelfand Transform Γ: A −→ C0(X) a 7→ ba where ba : X → C is the point evaluation ba(γ) = γ(a). 7/41 In particular, this means that X ' Y if and only if C0(X) 'C0(Y ). ∗ Every C -algebra homomorphism F : C0(Y ) → C0(X) is the pullback map of a continuous (and proper) map f : X → Y.