Fall 2008
The operator algebras seminar will meet on random Wednesdays during the semester at 2:10
in room 959.
29 October. Ilan Hirshberg: "The Jiang-Su algebra does not always embed"
Spring 2008
Click
here to download notes for a lecture on operator spaces (corrected 13 May, 2008).
Fall 2007
Mathematics 290: Operator Algebras Seminar
Summer schedule: Tuesdays, 2--3:30, Room 891 Evans.
7 August. Ilan Hirshberg: Izumi's work on contraction semigroups that
perturb the shift semigroup on L^2(0,\infty).
Recently, there has been important progress on the construction
and classification of semigroups of endomorphisms of B(H) by
Izumi, and Izumi-Srinivasan. These lectures will focus on results
of Izumi that play a central role in the above.
Click
here to download Izumi's preprint.
Click
here to download more details.
21 August. Ilan Hirshberg continues.
Fall Schedule: Wednesdays, 2--3, Room 961 Evans.
14 August. Ilan Hirshberg: Permutations of the Jiang-Su algebra.
19 September. Bill Arveson: Extremal marginal traces on matrix algebras
and quantum entanglement.
Click
here for more information.
26 September. Bill Arveson: Marginal traces and completely positive maps.
Click
here for more information.
Spring 2007
Mathematics 290: Operator Algebras Seminar
Wednesdays, 2--3, Room 961 Evans.
24 January. Bill Arveson: The noncommutative Choquet boundary.
Very recently, it was shown that every separable operator system
has sufficently many boundary representations.
This establishes the existence of the noncommutative
Choquet boundary in a rather general setting. I will
begin by giving a series of lectures on this
forty-year-old problem, describing its origins
in issues arising from operator theory,
the milestones along the way, and the promise
of further developments.
Click
here to download background material with references.
31 January. Bill Arveson: The noncommutative Choquet boundary II.
7 February. Bill Arveson: The noncommutative Choquet boundary III.
14 February. Bill Arveson: The noncommutative Choquet boundary IV.
We show that the unique extension property is hereditary in the
strongest possible sense: For every direct integral decomposition of
a UCP map with the unique extension property into UCP maps, almost
every one of the integrands has the unique extension property.
21 February. Bill Arveson: The noncommutative Choquet boundary V.
Continuation
28 February. Bill Arveson: The noncommutative Choquet boundary VI.
Completion of the proof of the main result. (hopefully).
4 April. Anders Hansen, Oslo university: pseudospectra of linear operators on Hilbert spaces.
11 April, 18 April. No meeting
25 April. Marius Junge, university of Illinois: Minimal sets for operator systems.
2 May. Robert Powers, university of Pennsylvania: Comparison theory for E0-semigroups.
Spring 2006
Mathematics 290: Operator Algebras Seminar
Wednesdays, 2--3, Room 939 Evans.
18 Jan. Bill Arveson: Survey of Noncommutative Dynamics I
I will begin by giving a general survey of the role of E_0-semigroups in
noncommutative dynamics. The initial lecture will focus on the differences
between the way the flow of time acts in probability theory versus quantum theory.
We discuss a natural notion of causality in the noncommutative setting, and show how
this leads naturally to the study of pairs of E_0-semigroups.
25 Jan. Bill Arveson: Survey of Noncommutative Dynamics II
We summarize some of the general theory of E_0-semigroups and
some of the central problems that remain unsolved -- including
product systems, the numerical
index, the role of cocycle perturbations, and the current state of
knowledge about classification. Subsequent lectures will focus on the properties
of {\em pure} E_0-semigroups - which can be viewed as noncommutative counterparts
of the so-called Kolmogorov endomorphisms of ergodic theory.
1 Feb. Continuation
8 Feb.
Click
here to download a pdf file containing lecture notes on continuous tensor products
of Hilbert spaces.
15 Feb. TBA
Fall 2005
Mathematics 290: Operator Algebras Seminar
Wednesdays, 2--4, Room 959 Evans.
We begin with a series of two or three lectures
by Ilan Hirshberg on self-absorbing C*-algebras.
First meeting: August 31.
Fall 2004
Mathematics 290: Operator Algebras Seminar
Wednesdays, 2--4, Room 939 Evans.
The central topic this semester will be
noncommutative Poisson boundaries. These are noncommutative
generalizations of the ``Poisson boundary" of the space
of all bounded harmonic functions on the open unit disk,
or on a domain in complex n-space or, more generally,
on a complete Riemannian manifold.
The noncommutative counterpart of the Laplacian is the
generator of a semigroup of completely positive maps acting
on a von Neumann algebra M, and the space H(M) of
noncommuative harmonic functions is by definition the
space of all elements of M that
are fixed under the action of the semigroup. H(M)
is an operator system; and while it is almost never closed
under the multiplication of M,
it has a new multiplication
with respect to which it is a von Neumann algebra.
This von Neumann algebra bH(M) is uniquely determined
by H(M) up to isomorphism.
bH(M) is the noncommutative Poisson boundary of H(M).
It is a central problem in the subject to identify
the structure of bH(M) in terms of the given data.
A week-by-week description of the seminar
follows. People following this seminar may wish
to download the lecture notes from the operator
algebras seminar held during Fall of 2003, posted below.
Lecture 1.
Lectures 2 and 3.
Lectures 4 and 5:
The range of an idempotent is a C*-algebra.
Noncommutative Poisson boundaries.
Fall 2003
Mathematics 290: Operator Algebras Seminar
Mondays 3:10--4, room 961 Evans
Ilan Hirshberg and I will discuss several topics
of current interest in operator theory/operator algebras.
Some knowledge of Hilbert space operators and
operator algebras will be assumed, but we will
provide background material when appropriate.
Click
here to download a pdf file containing updated
information about weekly topics.
I will occasionally post
links to downloadable pdf files
containing notes, historical comments
and references relating to various lectures below.
lecture 1 Introduction.
lecture 2:
The noncommutative Hahn-Banach Theorems.
lecture 3:
Extensions of C*-algebras.
lecture 4:
Extensions and liftings.
lecture 5:
The lifting theorem for nuclear C*-algebras.
Functional Analysis Colloquium.
Tuesdays 4:10--5, room 3 Evans
Meetings of the FAC are irregular, depending on the availability of
visitors, the phase of the moon, and other factors.
Click
here to download a pdf file containing the latest information.
Spring 2003
Mathematics 105: Analysis II
MWF 10, room 3109 Etcheverry hall
Text: "Real Mathematical Analysis"
by Charles Pugh,
Springer-Verlag Undergraduate Texts in Mathematics (2001)
Office Hours: MWF 11:10-12 in room 983 Evans hall.
See the department's course listing for information about other
courses.
The reader, Oscar Villareal, will be in his office,
Room 1093 Evans, on Fridays from
3--4, for consultation about problem sets.
Here are pdf-formatted files containing the problem sets.
TIP: Since these problems are being made up as we go,
you'd be wise to check each friday afternoon for updates/corrections.
Exercises due 3 February pdf.
Exercises due 19 February pdf.
Exercises due 24 February pdf.
Exercises due 3 March pdf.
Exercises due 10 March pdf.
Exercises due 31 March pdf.
Exercises due 7 April (corrected) pdf.
Exercises due 14 April pdf.
Exercises due 21 April pdf.
Exercises due 28 April pdf.
Lecture notes: Riemann integral vs. Lebesgue integral pdf.
Exercises due 5 May pdf.
Exercises due 12 May pdf.
The reader will put the graded 12 May assignment in the box
outside my office (983 Evans hall) sometime during Wednesday of this week.
They will be available on Thursday, 15 May.
Here are certain solution sets.
Problems due 19 February pdf.
Midterm exam pdf.
There will be a 50-minute midterm exam on Monday,
Mar 10, during the normal lecture period. The
exam will cover material discussed in the lectures
through friday, Mar 7. There will be no further
midterm exams, but there will be a final exam. Grades
will be determined by performance on 1) problem sets,
2) the midterm exam and 3) the final exam, according to
the following rough weighting:
Final: 35%
Problem sets: 35%
Midterm exam: 30%.
The FINAL EXAM is scheduled at the unfortunate hour of 8AM,
in room 9 Evans Hall, on Monday, May 19, 2003. You should
be sure to bring a blue book.
Graded problem sets that are not picked up in class
are available in a box outside my office anytime.
Today's trivia quiz
Who is this man? Here's a hint if you
need it.