K-theory of C*-algebras. Spring 2025
Syllabus
Warning. In Lectures 1-6, we give a very brief survey of Banach algebras and C*-algebras. For a more complete introduction,
see our earlier courses
"C*-algebras and compact quantum groups" (spring 2024)
and "Harmonic analysis and Banach algebras" (fall 2024).
Lectures
Playlist
- Lecture 1 (17.01.2025) (video)
- Historical remarks on K-theory. Banach algebras. Basic examples of Banach algebras (function algebras, operator algebras,
group L1-algebras of discrete groups). Unitization. The unitization of C0(X) and the one-point
compactification of X. The spectrum of a untial algebra element, examples. The nonunital spectrum. Relations between the unital and nonunital spectra.
Properties of the multiplicative group of a Banach algebra. The continuity of characters. The compactness and the nonemptiness of the spectrum.
The Gelfand-Mazur theorem.
- Lecture 2 (24.01.2025) (video)
- The behaviour of the spectrum under homomorphisms. Spectrally invariant subalgebras. The polynomial spectral mapping theorem.
The spectral radius formula. Maximal and modular ideals of a commutative algebra. The maximal spectrum and the character space of a commutative algebra.
The closedness of maximal modular ideals of a commutative Banach algebra.
The 1-1 correspondence between the character space and the maximal spectrum.
The Gelfand topology on the maximal spectrum. Topological properties of the maximal spectrum.
The Gelfand transform. An example: the maximal spectrum and the Gelfand transform for C0(X).
Functorial properties of the Gelfand transform (the adjoint functors X → C(X) and A → Max(A)).
- Lecture 3 (31.01.2025) (video)
- Banach *-algebras. C*-algebras. Examples. Unitizations of C*-algebras. Basic spectral properties of C*-algebras:
the spectral radius of a normal element is equal to the norm, the spectrum of a selfadjoint element is real.
Corollaries: the uniqueness of a C*-algebra norm, the automatic continuity of *-homomorphisms,
characters of C*-algebras preserve involution, the spectral invariance of C*-subalgebras.
The 1st (commutative) Gelfand-Naimark theorem. A category-theoretic interpretation of the Gelfand-Naimark theorem.
Remarks on noncommutative geometry. Injective *-homomorphisms are isometric.
- Lecture 4 (07.02.2025) (video)
- Continuous functional calculus in C*-algebras. The spectral mapping theorem and the superposition property for the functional calculus.
Positive elements in C*-algebras. Properties of positive elements. Square roots. Kaplansky's theorem (x*x≥0).
The order structure on a C*-algebra. Approximate identities in Banach algebras. Examples. The existence of
approximate identities in C*-algebras. Quotients of C*-algebras and related properties of *-homomorphisms.
- Lecture 5 (14.02.2025) (video)
- Positive functionals on a C*-algebra. Examples and basic properties of positive functionals. Positivity criteria for linear functionals.
Extension and existence of positive functionals. The GNS construction. Hilbert direct sums of *-representations. The universal representation
of a C*-algebra. The 2nd Gelfand-Naimark theorem (the existence of an isometric *-representation of a C*-algebra).
Tensor products of *-algebras. The C*-algebra structure on the matrix algebra Mn(A) (existence and uniqueness).
- Lecture 6 (21.02.2025) (video)
- Tensor products of Hilbert spaces, of operators on Hilbert spaces, and of *-representations. The spatial C*-norm and the spatial
tensor product of C*-algebras (several equivalent definitions). The functoriality of the spatial C*-tensor product.
Examples: the spatial tensor product of K(H) by K(H); the spatial tensor product by C0(X).
The maximal C*-tensor product. Properties of the maximal C*-tensor product. Nuclear C*-algebras. Examples.
- Lecture 7 (28.02.2025) (video)
- Projective modules over a unital ring: equivalent definitions. Free modules. A characterization of projective modules in terms of free modules.
The semigroup V(R) of isomorphism classes of finitely generated projective modules. Properties of V(R)
(functoriality, additivity, V(R)=V(Rop)). The Grothendieck group of an abelian semigroup:
universal property, construction, basic properties. The functor K0 for unital rings. An example:
K0 for principal ideal domains. K0 and stably isomorphic projective modules.
- Lecture 8 (07.03.2025) (video)
- Vector bundles and their morphisms. Some operations on vector bundles (Whitney sum, pullback). Sections of vector
bundles. The semigroup V(X) of isomorphism classes of vector bundles over X. The functor K0
for compact Hausdorff topological spaces. The Serre-Swan theorem. Corollaries: the isomorphisms
V(X)=V(C(X)) and K0(X)=K0(C(X)). Equivalence of idempotents in a ring.
The existence of reduced pairs implementing an equivalence of idempotents. A geometric model for
equivalence of idempotents.
- Lecture 9 (14.03.2025) (video)
- A description of V(R) in terms of equivalence classes of idempotents in the infinite matrix ring M∞(R).
Orthogonal idempotents and the semigroup structure on the set of equivalence classes of idempotents. The functorial behavior
of V(R) in terms of matrices. Examples: calculating K0 for the ring of linear operators on a vector space
(the finite-dimensional and the infinite-dimensional cases). Similarity of idempotents. Relations between similarity
and algebraic equivalence. A description of V(R) in terms of conjugacy classes of infinite idempotent matrices.
- Lecture 10 (21.03.2025) (video)
- Ring extensions. The unitization extension. The functor K0 for nonunital rings. A compatibility with the unital case.
The topological K0-functor for locally compact Hausdorff spaces. The isomorphism K0(X)=K0(C0(X)).
A characterization of K0(R) in terms of idempotents: the unital and the nonunital cases.
(Non)lifting invertibles under surjective ring homomorphisms: examples. Whitehead's lemma on lifting invertible matrices.
The half-exactness of K0.
References