Research

My research is at the intersection of functional  and applied harmonic analysis , in particular operator algebras, representation theory and frames (both theory and applications).

MOST RECENT  PROJECTS 

Frame theory and OVM — A dilation approach

This is a long term project with David Larson (Texas A&M University)  and Rui Liu (Nankai University) with the goal of classifying operator valued measures (both on commutative domain and non-commutative domain — projections in a von Neumann algebras) from the dilation point of view. This project was initially motivated by the dilations of frames, which can be considered as a special case of an OVM. There is corresponding theory for bounded linear maps. While the Hilbertian dilations usually associated with complete boundedness of the OVM or the linear map, Banach space theory plays an essential role in establishing a general dilation theory for arbitrary OVM/maps. The theory of quantum measures can be viewed as a non-commutative frame theory since their domain spaces contains non-commutative objects such as quantum states (projections) from a non-commutative von Neumann algebra. A general dilation theory for quantum measures need to be established.

Frame theory in quantum information 

  • Phase retrievable frames: Characterize/construct various type of frames that can be used to perform phase-retrieval for various signal spaces, i.e., recover rank one projection (pure state) xx* from the magnitudes of the from coefficients
  • Representation frames: Investigate phase retrieval property of the frames induced by group representations. An open problem is: Does every irreducible projective unitary representation of any finite non-abelian group admit a single vector such that its orbit is a phase-retrievable frame. While lots of work has been done for phase-retrievable Gabor frames, very little is known for general phase-retrievable  representation frames
  • Quantum channels: Examine various type of  quantum channels in connection with frame theory, in particular, covariant quantum channels, entanglement breaking  and mixed unitary channels.

Fourier Theory for fractal measures