Elliott Distinguished
Visitor Lectures
February 25  March 27
(video
of the talks)
Speaker:
Eberhard Kirchberg, HumboldtUniversität 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 idealsystem 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 WeylvonNeumann 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
bimodules, Cone ${\mathcal{C}}$dependent Extgroups
$\mathrm{Ext}(\mathcal{C};\, A,B)$, Related semigroups.

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 semigroups.

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 $\mathcal{O}_\infty$,
1step 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

Conerelated KKgroups 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

Conerelated KKgroups 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 idealsystem preserving zerohomotopy.

10

Some conclusions of the classication 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
idealequivariant classication, indications of possible
equivariant versions for actions of compact groups (up
to 2cocycle equivalence).

