SCIENTIFIC PROGRAMS AND ACTIVITIES

July 22, 2014
THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

January-June 2014
Thematic Program on Abstract Harmonic Analysis, Banach and Operator Algebras

March 2014
C*-Algebras and Dynamical Systems

Organizer: George Elliott

Program Outline
The six-month Thematic Program will begin with a month-long Winter School on Basics in Abstract Harmonic Analysis, Banach and Operator Algebras followed by four concentration periods, devoted to one of the Program themes.

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).
March 20-25 at 2:00 p.m.

Mini-session on "Multi-norms" (video of the talks)

March 20 & 21, H.G.Dales
Multi-norms 1, 2 (slides)

March 24, Vladimir Troitsky
Multi-norms and Banach lattices

March 25, Niels Laustsen
Ideals of operators on Banach spaces (slides)

March 26-28 at 3:30 p.m. von Neumann Lecture Series (video of the talks)
Uffe Haagerup, University of Copenhagen
Approximation Properties for Groups and von Neumann Algebras



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