This seminar, organized by Man-Duen Choi, Chandler Davis, George Elliott, and Peter Rosenthal, is devoted, loosely speaking, to the study of all areas of mathematics involving Hilbert space operators. (This has variously been interpreted recently to include matrix theory, index theory of differential operators and other aspects of non-commutative geometry, invariant subspace theory, quantum information theory, C*-algebra classification theory, and string theory.)

For a given orthonormal basis of a Hilbert space, a map from this basis (as a set) to itself can be extended as a (possibly unbounded) operator on the Hilbert space. We call such an operator a basis map operator, and classify these operators that arise as multiplications in the regular representation of a countable semigroup. We will review some of these results for various semigroup types (eg: finite, bounded, commutative, etc.), and discuss in greater detail the case of basis map operators arising from connected, but non-commutative semigroups. Finally we will end with an application of the above results to show that finitely generated, commutative, bounded semigroups must be also be uniformly bounded.

- · Monday, Jan. 25, 2010: K-theory of C*-algebras and KK-theory (Abhijnan Rej)
- · Monday, Feb. 01, 2010: Introduction to KK-theory (and its categorical properties) (Abhijnan Rej)
- · Monday, Feb. 08, 2010: The intersection product in KK-theory and non-commutative topology (Abhijnan Rej)
- · Monday, Feb. 22, 2010: D-branes and KK groups (Abhijnan Rej)
- · Monday, Mar. 22, 2010: Relative determinants and isospectrality on surfaces with cusps (Clara Aldana)
- · Monday, Mar. 29, 2010: The Cuntz semigroup of C(X,A) where A is a simple AF C*-algebra (Aaron Tikuisis)
- · Monday, Apr. 05, 2010: Basis map operators arising from regular semigroup representations (Barry Rowe)
- · Monday, Apr. 19, 2010: On the classification of C*-algebras (Henning Petzka)
- · Monday, May. 10, 2010: The Dirac operator on the noncommutative space of connections (Alan Lai)
- · Tuesday, May. 11, 2010: Naimark extensions for indeterminacy in the moment problem (Franek Szafraniec)
- · Monday, May. 31, 2010: On triangular factorization of positive operators (Lev A. Sakhnovich)
- · Monday, Sep. 20, 2010: Quantum error correction, and higher rank numerical ranges of normal matrices (Chi-Kwong Li)
- · Friday, Jan. 14, 2011: Spectral zeta functions and Gauss-Bonnet type theorems in noncommutative geometry (Masoud Khalkhali)
- · Monday, Feb. 07, 2011: Group actions and noncommutative geometry (Raphael Ponge )
- · Monday, Mar. 07, 2011: On certain multiplier projections (Henning Petzka)
- · Monday, Mar. 14, 2011: What are quantum channels through my old dream? (Man-Duen Choi)
- · Monday, Mar. 21, 2011: The nuclear dimension of a C* algebra (Nicola Watson)
- · Monday, Mar. 28, 2011: A generalization of the Weyl-von Neumann Theorem (Daniel Hay)
- · Monday, Mar. 19, 2012: Characterizing $\mathcal{Z}$-Stable $C^*$-algebras (Danny Hay)
- · Monday, Mar. 26, 2012: A stably finite $C^*$-algebra without bounded trace (Henning Petzka)
- · Monday, Apr. 02, 2012: Decomposition rank, nuclear dimension and tracial approximation (Nicola Watson)
- · Monday, Apr. 23, 2012: Hilbert Modules, Strong Morita Equivalence and Rotation Algebras (James Lutley)
- · Monday, May. 28, 2012: Random Projections, Free Probability, and Holomorphic PDEs (Todd Kemp)
- · Tuesday, Nov. 20, 2012: Universal C*-algebras of *-semigroups and the C*-algebra of a partial isometry (Berndt Brenken)
- · Monday, Dec. 17, 2012: Constructing subfactors with jellyfish (Dave Penneys)