|
Elliott Distinguished
Visitor Lectures
February 25 - March 27
(video
of the talks)
Speaker:
Eberhard Kirchberg, Humboldt-Universität zu Berlin
*please contact us at thematic<at>fields.utoronto.ca
if you notice any errors in the Lecture notes, thank you*
|
|
Content of Lectures |
|
1
|
Towards ideal-system equivariant classification
(Lecture
Notes)
Basic definitions and terminology, statement
of the main results: Embedding Theorem, Theorem on realization
of ${\mathrm{KK}(\mathcal{C}; \cdot,\cdot)}$) by C*-morphisms,
On applications.
|
|
2
|
An ideal system equivariant Embedding
Theorem (I) (Lecture
Notes)
A generalized Weyl-von-Neumann Theorem in
the spirit of Voiculescu and Kasparov, Actions of topological
spaces on C*-algebras versus matrix operator convex
cones $\mathcal{C}$, related "universal" Hilbert
bi-modules, Cone ${\mathcal{C}}$-dependent Ext-groups
$\mathrm{Ext}(\mathcal{C};\, A,B)$, Related semi-groups.
|
|
3
|
Ideal system equivariant Embedding Theorem (II)
(Lecture
Notes)
C*-systems and its use for embedding results,
the example of embeddings into $\mathcal{O}_2$, criteria
for existence of ideal equivariant liftings, i.e. characterization
of invertible elements in the extension semigroup.
|
|
4
|
Ideal system equivariant Embedding Theorem
(III)
Proof of a special case by construction of
a suitable C*-system, Outline of the idea for the proof
of the general case by study of asymptotic embeddings,
using continuous versions of Rørdam semi-groups.
|
|
5
|
Some properties of strongly purely infinite
algebras
Operations on the class of s.p.i. algebras,
coronas and asymptotic algebras of strongly purely infinite
algebras, tensorial absorption of $\mathcal{O}_\infty$,
1-step innerness of residually nuclear c.p. maps.
|
|
6
|
Rørdam groups R($\mathcal{C};\,
A,B)$ (I)
Definition and properties of the natural group
epimorphism from the $\mathcal{C}$-dependent Rørdam
group R($\mathcal{C};\, A,B)$ onto Ext($\mathcal{SC};
A; SB$), reduction of the isomorphism problem to the
question on homotopy invariance of R($\mathcal{C};\,
A,B)$, Some cases of automatic homotopy invariance:
the "absorbing" zero element.
|
|
7
|
Rørdam groups (II)
Homotopy invariance of R($\mathcal{C};\, A,B)$,
existence of C*-morphisms $\varphi:A \rightarrow B$
that represent the elements of R($\mathcal{C};\, A,B)$,
proof of the Embedding Theorem in full generality.
|
|
8
|
Cone-related KK-groups KK($\mathcal{C};\,
A,B)$) (I)
Definition and basic properties of $\mathcal{C}$-related
($\mathbb{Z_2}$-graded) Kasparov groups KK($\mathcal{C};\,
A,B$) for graded m.o.c. cones $\mathcal{C}$, the isomorphisms
Ext($\mathcal{C};\, A,B$) $\cong$ KK($\mathcal{C_{(1)}};\,
A,B_{(1)}$) and Ext($\mathcal{SC};\, A,SB$) $\cong$
KK($\mathcal{C};\, A,B$) in trivially graded case. Homotopy
invariance of Ext($\mathcal{SC};\, A,SB$). The isomorphism
Ext($\mathcal{SC};\, A,SB$) $\cong$ R($\mathcal{C};\,
A,B$).
|
|
9
|
Cone-related KK-groups KK($\mathcal{C};\,
A,B$) (II)
The $KK_{X}(A;B)$ := KK($\mathcal{C_{X}};\,
A,B$) classification for X $\cong$ Prim(A) $\cong$ Prim(B),
where A, B are stable amenable separable C*-algebras.Structure
of the algebras with ideal-system preserving zero-homotopy.
|
|
10
|
Some conclusions of the classi cation results and open questions
(Lecture Notes)
Constructions of examples of algebras with
given second countable locally compact sober $T_0$ spaces
(not necessarily Hausdorff). Minimal requirement for
a weak version of a universal coefficent theorem for
ideal-equivariant classi cation, indications of possible
equivariant versions for actions of compact groups (up
to 2-cocycle equivalence).
|
|
