Classification of Continuous Fields of C* Algebras Prahlad Vaidyanathan (Joint work with M. Dadarlat) Department of Mathematics IISER Bhopal Sept 11, 2014 2 Continuous Fields of C* Algebras 3 E-theory for Continuous fields 4 Classification Results Outline 1 Elliott’s Classification Program 3 E-theory for Continuous fields 4 Classification Results Outline 1 Elliott’s Classification Program 2 Continuous Fields of C* Algebras 4 Classification Results Outline 1 Elliott’s Classification Program 2 Continuous Fields of C* Algebras 3 E-theory for Continuous fields Outline 1 Elliott’s Classification Program 2 Continuous Fields of C* Algebras 3 E-theory for Continuous fields 4 Classification Results A finite dimensional C* algebra is of the form Mn1 (C) ⊕ Mn2 (C) ⊕ ::: ⊕ Mnk (C) A nuclear C* algebra is one that can be approximated “nicely" by finite dimensional C* algebras. Elliott’s Classification Program C* algebras A C* algebra is An algebra of bounded linear operators on a Hilbert space that is closed in the norm topology and closed under taking adjoints. A complex Banach algebra with an involution map a 7! a∗ such that ka∗ak = kak2 A nuclear C* algebra is one that can be approximated “nicely" by finite dimensional C* algebras. Elliott’s Classification Program C* algebras A C* algebra is An algebra of bounded linear operators on a Hilbert space that is closed in the norm topology and closed under taking adjoints. A complex Banach algebra with an involution map a 7! a∗ such that ka∗ak = kak2 A finite dimensional C* algebra is of the form Mn1 (C) ⊕ Mn2 (C) ⊕ ::: ⊕ Mnk (C) Elliott’s Classification Program C* algebras A C* algebra is An algebra of bounded linear operators on a Hilbert space that is closed in the norm topology and closed under taking adjoints. A complex Banach algebra with an involution map a 7! a∗ such that ka∗ak = kak2 A finite dimensional C* algebra is of the form Mn1 (C) ⊕ Mn2 (C) ⊕ ::: ⊕ Mnk (C) A nuclear C* algebra is one that can be approximated “nicely" by finite dimensional C* algebras. The Elliott Program To classify nuclear C* algebras by their K-theory Elliott’s Classification Program K-theory To each C* algebra A, one can associate two groups K0(A) and K1(A) Elliott’s Classification Program K-theory To each C* algebra A, one can associate two groups K0(A) and K1(A) The Elliott Program To classify nuclear C* algebras by their K-theory A class of simple algebras for which the Elliott program has been successful is the class of Kirchberg algebras (Purely infinite, simple, unital, nuclear algebras) Fact Every separable nuclear C* algebra is K-theoretically equivalent to a Kirchberg algebra. Elliott’s Classification Program Kirchberg Algebras A C* algebra is called simple if it has no non-trivial closed two-sided ideals. Fact Every separable nuclear C* algebra is K-theoretically equivalent to a Kirchberg algebra. Elliott’s Classification Program Kirchberg Algebras A C* algebra is called simple if it has no non-trivial closed two-sided ideals. A class of simple algebras for which the Elliott program has been successful is the class of Kirchberg algebras (Purely infinite, simple, unital, nuclear algebras) Elliott’s Classification Program Kirchberg Algebras A C* algebra is called simple if it has no non-trivial closed two-sided ideals. A class of simple algebras for which the Elliott program has been successful is the class of Kirchberg algebras (Purely infinite, simple, unital, nuclear algebras) Fact Every separable nuclear C* algebra is K-theoretically equivalent to a Kirchberg algebra. By the fact mentioned above, we may assume that all simple quotients of our C* algebras are Kirchberg algebras. This Project Goal To classify non-simple C* algebras using K-theory K-theory of all quotients Maps between these K-theory groups This Project Goal To classify non-simple C* algebras using K-theory K-theory of all quotients Maps between these K-theory groups By the fact mentioned above, we may assume that all simple quotients of our C* algebras are Kirchberg algebras. Θ is called the structure morphism. We usually drop Θ and write f · a := Θ(f )(a) Continuous Fields of C* Algebras C(X)-algebras Let X be a compact metric space. A structure of a C(X)-algebra on a separable C* algebra A is a unital *-homomorphism Θ: C(X) !ZM(A) from C(X) to the center of the multiplier algebra of A. In other words, A has the structure of a non-degenerate C(X)-module. Continuous Fields of C* Algebras C(X)-algebras Let X be a compact metric space. A structure of a C(X)-algebra on a separable C* algebra A is a unital *-homomorphism Θ: C(X) !ZM(A) from C(X) to the center of the multiplier algebra of A. In other words, A has the structure of a non-degenerate C(X)-module. Θ is called the structure morphism. We usually drop Θ and write f · a := Θ(f )(a) Write πx : A ! A(x) for the natural quotient map. For a 2 A, write a(x) := πx (a) A C(X)-algebra A is called a continuous field of C* algebras over X if the function x 7! ka(x)k is continuous on X for each a 2 A Continuous Fields of C* Algebras Definition and Examples Continuous Fields of C* Algebras The C(X)-module structure allows us to localize at points of X. For x 2 X, the quotient A A(x) := C0(X n fxg) · A is called the fiber of A at x. Continuous Fields of C* Algebras Definition and Examples Continuous Fields of C* Algebras The C(X)-module structure allows us to localize at points of X. For x 2 X, the quotient A A(x) := C0(X n fxg) · A is called the fiber of A at x. Write πx : A ! A(x) for the natural quotient map. For a 2 A, write a(x) := πx (a) A C(X)-algebra A is called a continuous field of C* algebras over X if the function x 7! ka(x)k is continuous on X for each a 2 A If X = Prim(A) Hausdorff, the Dauns-Hoffman Theorem says that ∼ C(X) = ZM(A) and A(x) is simple for all x 2 X. Thus, A unital separable C* algebra with a Hausdorff primitive spectrum is a continuous field over its spectrum. Continuous Fields of C* Algebras Definition and Examples The Primitive Spectrum If A is a C* algebra A primitive ideal is the kernel of an irreducible representation. The collection of primitive ideals, Prim(A), is a compact T0-space Thus, A unital separable C* algebra with a Hausdorff primitive spectrum is a continuous field over its spectrum. Continuous Fields of C* Algebras Definition and Examples The Primitive Spectrum If A is a C* algebra A primitive ideal is the kernel of an irreducible representation. The collection of primitive ideals, Prim(A), is a compact T0-space If X = Prim(A) Hausdorff, the Dauns-Hoffman Theorem says that ∼ C(X) = ZM(A) and A(x) is simple for all x 2 X. Continuous Fields of C* Algebras Definition and Examples The Primitive Spectrum If A is a C* algebra A primitive ideal is the kernel of an irreducible representation. The collection of primitive ideals, Prim(A), is a compact T0-space If X = Prim(A) Hausdorff, the Dauns-Hoffman Theorem says that ∼ C(X) = ZM(A) and A(x) is simple for all x 2 X. Thus, A unital separable C* algebra with a Hausdorff primitive spectrum is a continuous field over its spectrum. If A is a continuous field of C* algebras such that, for each x 2 X, there is a neighbourhood V of x and a C* algebra D such that ∼ C0(V ) · A = C0(V ; D) then A is said to be locally trivial. Locally trivial fields are rare when the fibers are infinite dimensional C* algebras. Continuous Fields of C* Algebras Definition and Examples Examples of Continuous Fields If D is any fixed C* algebra, then A = C(X; D) = ff : X ! D such that f is continuous g is called the trivial field with fiber D Continuous Fields of C* Algebras Definition and Examples Examples of Continuous Fields If D is any fixed C* algebra, then A = C(X; D) = ff : X ! D such that f is continuous g is called the trivial field with fiber D If A is a continuous field of C* algebras such that, for each x 2 X, there is a neighbourhood V of x and a C* algebra D such that ∼ C0(V ) · A = C0(V ; D) then A is said to be locally trivial. Locally trivial fields are rare when the fibers are infinite dimensional C* algebras. Continuous Fields of C* Algebras Definition and Examples A field with one singularity Let D; H be two C* algebras, and γ : D ! H an injective *-homomorphism. For x 2 [0; 1] fixed, define Bx := f(f ; g) 2 C([0; x]; H) ⊕ C([x; 1]; D): f (r) = γ(g(x))g We represent this pictorially by H •0 •x O γ D •x •1 Then, Bx is a continuous field over [0; 1] which is locally trivial at all points of [0; 1] except x. Continuous Fields of C* Algebras Definition and Examples Elementary fields over [0; 1] An elementary continuous field over [0; 1] is a field which is locally trivial at all but finitely many points. We depict them pictorially by H1 H2 H3 Hk •0 •x1 •x2 •x3 •x4 ::: •xn •1 O O O O O γ1 γ2 γ3 γ4 γn D1 D2 Dk−1 •x1 •x2 •x3 •x4 ::: •xn Question Given two continuous fields A and B over the same space X, how do we know if there is a C(X)-linear *-isomorphism ∼ ' : A −! B Continuous Fields of C* Algebras The Classification Problem Let A and B be a continuous fields over a space X.