C*-algebras and compact quantum groups. Spring 2021
Syllabus
- Lecture 1 (14.01.2021) (video)
(lecture notes)
- 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 (21.01.2021) (video)
(lecture notes)
- 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 (28.01.2021) (video)
(lecture notes)
- 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 spectrum of an element of a nonunital algebra. Modular ideals.
The maximal spectrum and the Gelfand transform for nonunital Banach algebras.
- Lecture 4 (04.02.2021) (video)
(lecture notes)
- Banach *-algebras. C*-algebras. Examples. Products and 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.
- Lecture 5 (11.02.2021) (video)
(lecture notes)
- 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.
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.
- Lecture 6 (18.02.2021) (video)
(lecture notes)
- Positive elements in C*-algebras. Properties of positive elements.
Square roots. Kaplansky's theorem (x*x≥0). Characterizations of positivity. The order structure on a C*-algebra.
Approximate identities in Banach algebras. Examples.
- Lecture 7 (25.02.2021) (video)
(lecture notes)
- The existence of approximate identities in C*-algebras. Quotient C*-algebras.
Positive functionals on a C*-algebra. Examples and basic properties of positive functionals.
- Lecture 8 (04.03.2021) (video)
(lecture notes)
- 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 9 (11.03.2021) (video)
(lecture notes)
- 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.
- Lecture 10 (18.03.2021) (video)
(lecture notes)
- GNS representations and cyclic vectors. Complex conjugate spaces. Tensor products of algebras and of *-algebras.
The C*-algebra structure on the matrix algebra Mn(A) (existence and uniqueness).
- Lecture 11 (25.03.2021) (video)
(lecture notes)
- 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). The functoriality of the spatial C*-tensor product.
- Lecture 12 (01.04.2021) (video)
(lecture notes)
- 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).
The maximal C*-tensor product.
- Lecture 13 (08.04.2021) (video)
(lecture notes)
- Properties of the maximal C*-tensor product. Nuclear C*-algebras. Examples. Remarks on tensor products
of locally convex spaces and on nuclear spaces. C*-envelopes and universal C*-algebras.
Sufficient conditions for the existence of C*-envelopes.
- Lecture 14 (15.04.2021) (video)
(lecture notes)
- Examples of C*-envelopes. Coalgebras, bialgebras, unital C*-bialgebras. A characterization
of compact groups as compact semigroups with cancellation. Compact quantum groups.
The Gelfand-Naimark-Woronowicz duality.
- Lecture 15 (22.04.2021) (video)
(lecture notes)
- The universal property of C(SU(2)). The quantum SU(2) group.
Compact matrix quantum groups. The quantum SU(2) as the C*-envelope of the quantum SL(2).
- Lecture 16 (29.04.2021) (video)
(lecture notes)
- Slice maps on spatial C*-tensor products. Left invariant and right invariant
functionals on compact quantum groups. The convolution product
on the dual space of a compact quantum group. Invariant functionals in terms of convolutions.
The existence and the uniqueness of a Haar state on a compact quantum group.
- Lecture 17 (13.05.2021) (video)
(lecture notes)
- The existence of the Haar state (end of proof). Finite-dimensional corepresentations of compact quantum groups.
Intertwining maps. An equivalence between representations of G and corepresentations of C(G).
Matrix corepresentations. The fundamental corepresentation of a compact matrix quantum group.
- Lecture 18 (20.05.2021) (video)
(lecture notes)
- The embedding of the category of corepresentations of a compact quantum group A into the category of representations
of A*. Invariant subspaces for corepresentations; the role of the dual algebra. Irreducble corepresentations.
Schur's lemma. Finite-dimensional corepresentations of compact quantum groups are unitarizable. A corollary:
compact matrix quantum groups are unitarizable. A characterization of commutative compact matrix quantum groups.
Philosophy: compact matrix quantum groups = compact quantum Lie groups. Concluding remarks
(decomposing into irreducibles, the dense Hopf *-subalgebra of matrix elements, Tannaka-Krein duality).
References