Operator Algebras
Operator Theory
Functional Analysis
Matrix Analysis
My work develops index theory beyond B(H). Classical Fredholm theory lives on a Hilbert space, where the compact operators are the distinguished ideal. Replacing them by a general separable stable C*-algebra B replaces B(H) by the multiplier algebra M(B), and the classical integer invariants become K₀(B)-valued. My research follows five directions:
Essential codimension and spectral flow. I extended the Brown–Douglas–Fillmore essential codimension to projections in M(B), gave both a concrete and a KK-theoretic definition and proved they agree, and used the invariant to characterize when projections lift from corona algebras whose ideals need not have real rank zero. With P. W. Ng and C. Wang I developed a spectral flow theory for paths of self-adjoint Fredholm operators in M(B), valid even for stably projectionless algebras, including a spectral flow isomorphism.
Commutators in multiplier algebras. Brown and Pearcy characterized commutators in B(H) in 1965; the analogous question for M(B) is open, and is the problem organizing this direction. Related work on K₁-injectivity of the Paschke dual algebra, multiplier unitary groups, and nilpotent perturbations supplies the tools.
Commutators in rings and matrix rings. With Arijit Mukherjee and Gobinda Sau I proved that a single commutator relation, together with a conjugation twisting it by a root of unity, forces a ring to decompose as a matrix ring over one of its corners.
Purely infinite rings. I am studying whether the C*-completion of a simple purely infinite ring is again simple purely infinite, under the hypothesis that the completion has real rank zero.
Completely positive maps and C-extreme points.* With Anand O. R. and K. Sumesh I studied C*-extreme points of the C*-convex set of unital completely positive maps on real C*-algebras, obtaining an extremality criterion that differs from the complex case.
I am also interested in noncommutative function theory — sum-of-Hermitian-squares extensions of noncommutative polynomials, and connections to real algebraic geometry, semidefinite programming, and optimization — and in commuting Toeplitz operators on Fock spaces.