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.