C*-algebras and compact quantum groups. Spring 2015

A. Yu. Pirkovskii

Syllabus

Lecture 1 (22.01.2015). 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). Unitization. The unitization of C₀(X) and the one-point compactification of X.

Lecture 2 (29.01.2015). The spectrum of an algebra element. Algebraic properties of the spectrum. The spectral mapping theorem for polynomials. The nonunital spectrum. Properties of the multiplicative group of a Banach algebra. The compactness and the nonemptiness of the spectrum. The Gelfand-Mazur theorem. The spectral radius. Maximal ideals in commutative algebras.

Lecture 3 (05.02.2015). The maximal spectrum and the character space of a unital commutative algebra. The automatic continuity of characters of a Banach algebra. The closedness of maximal ideals of a unital Banach algebra. The 1-1 correspondence between the character space and the maximal spectrum of a unital commutative Banach algebra. Remarks on locally convex topologies. 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. Maximal ideals in C(X).

Lecture 4 (12.02.2015). The Gelfand transform of "concrete" algebras: (1) The maximal spectrum and the Gelfand transform for "large" subalgebras of C(X); (2) The Gelfand spectrum of C_b(X) and the Stone-Cech compactification; (3) The Pontryagin dual and the Fourier transform of discrete abelian groups. Functorial properties of the Gelfand transform (the adjoint functors X -> C(X) and A -> Max(A)). The maximal spectrum and the Gelfand transform for nonunital Banach algebras.

Lecture 5 (19.02.2015). Banach *-algebras. C*-algebras. Examples: algebras of continuous functions, algebras of operators, reduced group C*-algebras of discrete groups, the irrational rotation algebra, the Toeplitz algebra. Products and unitizations of C*-algebras.

Lecture 6 (26.02.2015). Spectra of unitary and selfadjoint elements of a C*-algebra. The spectral radius of a normal element. Spectral invariance of C*-subalgebras. The 1st (commutative) Gelfand-Naimark theorem. A category-theoretic interpretation of the Gelfand-Naimark theorem. Properties of *-homomorphisms between C*-algebras.

Lecture 7 (05.03.2015). The continuous functional calculus in C*-algebras. Positive elements and the order structure on a C*-algebra. Convergence of nets in topological spaces.

Lecture 8 (12.03.2015). Approximate identities in normed algebras. The existence of approximate identities in C*-algebras. Quotient C*-algebras. Positive functionals on a C*-algebra.

Lecture 9 (19.03.2015). Positive functionals on a C*-algebra. Criteria of positivity. Extension of positive functionals. Representations of C*-algebras. Decomposing nondegenerate representations into cyclic summands.

Lecture 10 (02.04.2015). The GNS construction. An abstract characterization of GNS representations. GNS representations and cyclic vectors. The 2nd Gelfand-Naimark theorem (the existence of an isometric *-representation of a C*-algebra).

Lecture 11 (09.04.2015). Tensor products of *-algebras, 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 minimal tensor product.

Lecture 12 (16.04.2015). 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 C_0(X). Positive maps between C*-algebras and their continuity. Representations of tensor products of C*-algebras. The maximal C*-norm and the maximal tensor product of C*-algebras.

Lecture 13 (23.04.2015). The maximal tensor product of C*-algebras. Nuclear C*-algebras. The nuclearity of directed unions of C*-algebras. The nuclearity of K(H). Remarks on tensor products of Banach and locally convex spaces. Hilbert C*-modules. Basic examples. Summable families in normed spaces. Hilbert direct sums of C*-modules.

Lecture 14 (30.04.2015). Morphisms and adjointable operators between Hilbert C*-modules. Characterizations of selfadjoint and positive adjointable operators. Compact operators between Hilbert C*-modules. Tensor products of Hilbert C*-modules and Hilbert spaces.

Lecture 15 (14.05.2015). Tensor products of operators between Hilbert spaces and Hilbert modules. Multiplier algebras. Multiplier algebras of C*-algebras as algebras of adjointable operators. The extension theorem for *-representations of C*-algebras on Hilbert modules. The universal property of multiplier algebras.

Lecture 16 (21.05.2015). The multiplier algebra as the maximal essential extension. The multiplier algebras of K_A(E), K(H), and C_0(X). C*-algebra morphisms and the nonunital Gelfand-Naimark duality. Coalgebras, bialgebras, C*-bialgebras. A characterization of compact groups as compact semigroups with cancellation. Compact quantum groups. The Gelfand-Naimark-Woronowicz duality between compact groups and commutative compact quantum groups.

Lecture 17 (28.05.2015). An example: the reduced group C*-algebra of a discrete group is a compact quantum group. C*-envelopes and universal C*-algebras. Existence of C*-envelopes, examples of C*-envelopes. Compact matrix quantum groups. An example: the algebra of continuous functions on a closed subgroup of U(n) is a compact matrix quantum group.

Lecture 18 (11.06.2015). The quantum SU(2) group. Relations between (the algebra of regular functions on) the quantum SL(2) and (the algebra of continuous function on) the quantum SU(2). The quantum SU(n), the free orthogonal and the free unitary quantum groups, the free quantum permutation group. Slice maps on C*-algebras.

Lecture 19 (17.06.2015). Basic facts on the Haar measure on locally compact groups. 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 20 (19.06.2015). Basic facts on representations of topological groups. Unitary comodules and corepresentations of compact quantum groups. Matrix corepresentations. The right regular corepresentation. A survey of corepresentation theory (invariant subspaces, irreducible corepresentations, Schur's lemma, the invariance of orthogonal complements, the averaging procedure, the unitarizability of corepresentations, decomposing unitary corepresentations into finite-dimensional irreducibles, the dense Hopf *-algebra of matrix coefficients.)

• Take-home exam