### A one-sentence vita

I studied mathematics at CalTech and UCLA
and have held faculty positions at UCLA (sp 65),
Harvard (65-68), and Berkeley (68--)
where I've been professor of mathematics since 1974, with various visiting
positions and fellowships including Newcastle (UK), Aarhus, Rio de Janeiro, Oslo,
UCSD, Nankai, Canberra, Penn, Trondheim,
Kyoto, two years (1985-86 and 1999-00) as
Miller research professor at Berkeley, and possibly others I've forgotten.

I've also served on the editorial staff of several journals, including Bull. AMS,
Duke Math. J. in past years, and currently
JOT and
BJMJ .

If you MUST have details, here is a fairly current
CV.

### Research interests

I'm interested in "noncommutative analysis", which has
meant various things to me at different moments of my mathematical life.
It usually involves the use of operator algebras *per se*, or at least
the approach and philosophy of operator algebras, to get at problems
arising often outside of operator algebras. These problems have
involved a number of things that impressed me as interesting
and important over the years, including invariant subspaces,
the relation between function theory and operator
algebras, dynamical groups, stochastic nonlinear filtering,
the Feynman-Kac formula, dynamical properties of quantum fields,
numerical computation of spectra (of self-adjoint operators), and the
connections between multivariable operator theory and commutative algebra.
More recently, I've returned to the study of completely positive maps
of von Neumann algebras, and have been investigating the connection
between entanglement of states in quantum information theory and
the structure of completely positive maps on matrix algebras (yes,
the latter is finite-dimensional stuff).

During the mid 1980s I became convinced that it is important
to understand semigroups of endomorphisms of operator algebras
(E_0-semigroups). These objects arise naturally when one looks
carefully at the way the flow of time acts on the
algebra of observables of quantum theory. The flow of time in
quantum theory is fundamentally
different from the flow of time in classical
mechanics. In the Heisenberg picture, the flow of time
affects observables by causing them to change continuously
as time evolves. That much is true in both classical mechanics
and quantum mechanics.
The fundamental difference is that
in classical mechanics the algebra of observables is
the commutative algebra of functions on some space,
while in quantum mechanics or quantum field theory
(or quantum gravity for that matter),
the observables are operators on a Hilbert space.
The algebra of operators on a Hilbert space
is noncommutative.

It is this noncommutativity
of operators on a Hilbert space that provides a precise
formulation of the uncertainty principle:
there are operator solutions to equations like
pq - qp = 1. This equation has no commutative counterpart.
In fact, it has no solution in operators p,q acting on
a finite dimensional space.

So if you're interested in the dynamics of quantum
theory, you **must** work with operators
rather than functions and, more precisely,
operators on infinite
dimensional spaces.

E_0-semigroups are related to the theory of subfactors, but there
are important differences stemming from the fact that we are
dealing here with a continuous time parameter.

Much more on this subject
can be found in my monograph on noncommutative
dynamics, see the "Books, research papers,...." link that can be
accessed from my
home page.

More recently,
my efforts to understand noncommutative dynamics have led me
to look more closely at the theory of n-tuples of operators
acting on a Hilbert space (I won't explain why in these remarks).
In an attempt to lay hands on computable numerical
invariants, I discovered that there is a "curvature" invariant that
can be associated with certain commuting n-tuples of Hilbert space
operators.

This numerical invariant is defined as the integral over
the (2n-1)-sphere of the trace of a certain operator-valued function, and
is loosely analogous to the average Gaussian curvature of Riemannian
geometry. Like the geometric invariant, it turns out to be an
integer - but *what* integer? The answer appears to be
that the curvature invariant is the index
of a certain "Dirac" operator that is associated with the n-tuple
of operators. This is analogous to the current reformulation of
the Gauss-Bonnet-Chern formula as an index theorem, but it is
based on very different operator-theoretic considerations.

Completely positive maps on von Neumann algebras or between C*-algebras have
fascinated me since my days as a graduate student. Just when I think I
have finally understood them, something new comes along that shows me I
haven't. Completely positive maps on noncommutative algebras arise in many
contexts, including the dynamics of irreversible quantum systems, spatial
realizations of von Neumann algebras, probabilistic issues associated with
quantum theory in general, and in particular
quantum computing and quantum information theory.

What goes around has come around, and today quantum
information theory has led us back
into a finite-dimensional context. Completely positive maps on matrix algebras
are the objects that
are dual to quantum channels; in fact, the study of quantum channels reduces
to the study of completely positive maps of matrix algebras
that preserve the unit. This is an area that is still undergoing
vigorous development in efforts to understand entanglement, entropy,
and channel capacity in QIT.