C*-algebras and compact quantum groups. Spring 2024
Syllabus
- Lecture 1 (19.01.2024) (video)
- Historical remarks on C*-algebras and compact quantum groups.
Banach algebras. Basic examples of Banach algebras (function algebras, operator algebras, group L1-algebras of
discrete groups). A survey of spectral theory in Banach algebras (the spectrum
of an algebra element, algebraic properties of the spectrum, 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 (26.01.2024) (video)
- The spectral radius formula. The maximal spectrum and the character space of a commutative unital algebra.
The closedness of maximal ideals of a commutative unital Banach algebra.
The 1-1 correspondence between the character space and the maximal spectrum of a commutative unital Banach algebra.
Some properties of the weak* topology.
The Gelfand topology on the maximal spectrum. The compactness of the maximal spectrum (the unital case).
The Gelfand transform of a commutative unital Banach algebra. Remarks on the Jacobson radical.
- Lecture 3 (02.02.2024) (video)
- The maximal spectrum and the Gelfand transform for subalgebras of C(X). Functorial properties of the Gelfand transform
(the adjoint functors X → C(X) and A → Max(A)). Unitization. The unitization of C0(X) and the one-point
compactification of X. The nonunital spectrum of an algebra element. Relations between the unital and nonunital spectra.
- Lecture 4 (09.02.2024) (video)
- Modular ideals. The maximal spectrum and the Gelfand transform for nonunital Banach algebras. Banach *-algebras. C*-algebras.
Examples. Products and unitizations of C*-algebras.
- Lecture 5 (16.02.2024) (video)
- Unitizations of C*-algebras. The spectral radius of a normal element is equal to the norm. The automatic continuity of *-homomorphisms to C*-algebras.
The spectrum of a selfadjoint element is real. 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.
- Lecture 6 (01.03.2024) (video)
- Injective *-homomorphisms between C*-algebras are isometric. The 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.
- Lecture 7 (15.03.2024) (video)
- Kaplansky's theorem (x*x≥0). Characterizations of positivity. The order structure on a C*-algebra.
Approximate identities in Banach algebras. Examples. The existence of approximate identities in C*-algebras.
- Lecture 8 (22.03.2024) (video)
- The existence of approximate identities in C*-algebras (cont'd). Quotient C*-algebras.
Positive functionals on a C*-algebra. Examples and basic properties of positive functionals.
- Lecture 9 (29.03.2024) (video)
- Positivity criteria for linear functionals on a C*-algebra. Extension and existence of positive functionals.
*-representations and *-modules. The GNS construction. An abstract characterization of GNS representations.
- Lecture 10 (05.04.2024) (video)
- 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). Nondegenerate and cyclic *-representations.
Decomposing nondegenerate representations into cyclic summands. Cyclic representations = GNS representations.
- Lecture 11 (12.04.2024) (video)
- Tensor products of algebras and of *-algebras. The C*-algebra structure on the matrix algebra Mn(A)
(existence and uniqueness). 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. Takesaki's theorem on the minimality
of the spatial norm (without proof).
- Lecture 12 (19.04.2024) (video)
- The functoriality of the spatial C*-tensor product. The independence of the spatial C*-norm on the choice of faithful
representations. Examples: the spatial tensor product of K(H) by K(H); the spatial tensor product by C0(X).
- Lecture 13 (26.04.2024) (video)
- The maximal C*-tensor product. Properties of the maximal C*-tensor product. Nuclear C*-algebras. Examples.
C*-envelopes. Sufficient conditions for the existence of C*-envelopes.
- Lecture 14 (10.05.2024) (video)
- Universal C*-algebras defined by generators and relations. Examples. Coalgebras, bialgebras, unital C*-bialgebras. A characterization
of compact groups as compact semigroups with cancellation.
References