Operator Algebras Seminar, Spring 1997

• Date: 4/4/97
  • Speaker: Ed Effros, UCLA
  • Title: New ultraproduct techniques in operator algebra theory
  • Abstract: Although there is a natural embedding of a Banach space $V$ in its second dual $V^{**},$ the latter is generally much larger than the former. Nevertheless, as Grothendieck showed forty-five years ago, ``locally'' $V^{**}$ resembles $V$ in the sense that one can approximate finite dimensional subspaces of $V^{**}$ by finite dimensional subspaces of $V$. This fact is generally referred to as the ``local reflexivity'' property of Banach spaces, and has played a fundamental role in that theory. There is an obvious analogue of the notion of local reflexivity in the category of operator spaces and completely contractive mappings. Although a wide category of $C^{*}$-algebras indeed has this property, it is known that the analogue of this result is false in general. Turning to operator spaces which are not $C^{*}$-algebras, it seemed only natural to assume that an ``elementary'' example such as the operator space $\mathcal{S}_{1}$ of trace class operators must also be local reflexive. This has turned out to be surprisingly difficult to prove. The affirmative result is due to Junge, who used novel generalized ultraproduct methods to accomplish this. In this lecture we will present a simplified proof based on a characterization of local reflexivity due to Ruan and the speaker, together with Pisier's non-commmutative analogue of the Grothendieck-Pietsch theorem, and a key ultraproduct result of Junge.

• Date: 4/11/97
  • Speaker: Ken Goodearl, UCSB
  • Title: Ideals in Multiplier Algebras, I
  • Abstract: The multiplier algebra M(A) of a non-unital C*-algebra A is the largest (in a suitable sense) unital C*-algebra containing A as an ideal. For instance, if A is the algebra of continuous functions vanishing at infinity on a locally compact Hausdorff space X, then M(A) is the algebra of continuous functions on the Stone-Cech compactification of X. A general theme is that for typical A, the algebra M(A) is very "large" in many ways. For example, even when A is simple (i.e., has no nontrivial ideals), M(A) can have uncountably many ideals. This pair of talks will be very expository; I will introduce (almost) all the necessary background. My aim is to discuss the construction of M(A), describe various results about its ideal structure, and then develop -- from scratch -- just enough K-theory to be able to sketch how such results can be obtained. This part could also serve as propaganda and motivation for the K-theoretically-challenged.

• Date: 4/18/97
  • Speaker: Ken Goodearl, UCSB
  • Title: Ideals in Multiplier Algebras, II

• Date: 4/25/97
  • Speaker: Anne Louise Svendsen, UCSB
  • Title: The Pimsner-Popa inequality - a probabilistic description of the Jones index, Part I

• Date: 5/2/97
  • Speaker: Anne Louise Svendsen, UCSB
  • Title: The Pimsner-Popa inequality - a probabilistic description of the Jones index, Part II

• Date: 5/8/97
  • Speaker: Vaughan Jones, UC Berkeley
  • Title: Planar algebras
  • Abstract: Planar algebras are algebras whose elements can be represented by pictures in the plane on which various planar operations can be performed. They arise in knot theory but also in many other contexts. I will give many examples and state a theorem of Popa relating planar algebras to subfactors of von Neumann factors.

• Date: 5/9/97
  • Speaker: Peter Akemann, Treyarch Invention and UC Berkeley
  • Title: Subfactors and partially commuting squares

• Date: 5/16/97
  • Speaker: Chuck Akemann, UCSB
  • Title: Spectral scales of n-tuples in a II_1 factor
  • Abstract: Let $\{ b_1, \dots , b_n \}$ be an n-tuple of self adjoint elements in a II$_1$ factor M with trace tr. Define a map $P$ from M into $R^{n+1}$ by $P(a) = (tr(a), tr(ab_1), \dots, tr(ab_n))$. Let $B$ denote the image under $P$ of the positive part of the unit ball of M. We call this the {\it spectral scale} of the (n+1)-tuple $\{ tr, b_1, \dots, b_n \}$. The compact, convex set $B$ "determines" the (n+1)-tuple if the elements $\{b_1, \dots, b_n \}$ commute. If they don't commute, a matricial version is needed to completely "determine" the (n+1)-tuple. ("Determine" means up to unitary equivalence of the trace representation of the von Neumann subalgebra $N$ generated by $\{1, b_1, \dots, b_n \}$.)}

• Date: 5/23/97
  • Speaker: Dimitri Shlyakhtenko, UC Berkeley
  • Title: Free quasi-free states
  • Abstract: Free quasi-free states are free-probability analogs of the quasi-free states on the CAR and CCR algebras. Considering the von Neumann algebra arising in the GNS representation of a free quasi-free state leads one to free analogs of Araki-Woods factors. We discuss the properties of free quasi-free states, and some classification results on the free Araki-Woods factors

• Date: 5/30/97
  • Speaker: Masamichi Takesaki, UCLA
  • Title: The Characteristic Square for a Factor and Group Actions
  • Abstract: Each factor on a separable Hilbert space gives rise to a commutative square of exact sequences consisting of nine (3x3) middle non-trivial terms. The square is equivariant relative to the canonical action of $R \times Aut(M)$. We call it the characteristic square. This square gives then a corresponding cohomology element which will be called the intrinsic invariant of the factor. This element will give us a cocycle conjugacy invariant when it is pulled back by an action of a group. I will further discuss the meaning of the classification in analysis.