Hemant Pendharkar, Professor and Associate Chair in the Department of Mathematics & Statistics at the University of South Florida, recently co-authored a research article with Dr. Don Hadwin, Professor of Mathematics & Statistics at the University of New Hampshire, entitled "Central Sequences in Subhomogeneous Unital C* Algebras", published by Cambridge University Press. This and related work were carried out during the years 1994-99 and continues through today.
The main area of focus is in the area of operator algebras, specifically C* algebras (pronounced C-star); broadly in the area of mathematical analysis. These algebras were first considered primarily for their use in quantum mechanics to model algebras of physical observables.
You might be wondering though, what is an operator algebra and what is its connection to C* algebras? According to nLab, a webpage for the compiled and collaborative work on Mathematics, Physics, and Philosophy:
“An operator algebra is any subalgebra of the algebra of continuous linear operators on a topological vector space, with composition as the multiplication. In most cases, the space is a separable Hilbert space, and most attention historically has been paid to algebras of bounded linear operators. The operator algebras themselves are often equipped with their own topologies (e.g. norm topology, weak topology, weak* topology and so on) and sometimes involution. C*-algebras are (in the complex case) norm-closed complex *-subalgebras of the algebra of the bounded linear operators on a Hilbert space.”
This new paper deals with a specific subclass of C*algebras - Subhomogeneous Unital C*algebras, those whose representations have a bound. It gives an understanding of these algebras using the analysis of their central sequences.
“Consider how solutions to a quadratic equation are best understood when solved algebraically and simultaneously graphed to see where it crosses the x-axis. This reflective process is what helps us internalize mathematical concepts. In this paper, we similarly work with representations of these algebras and simultaneously work with central sequences that provide a reflective scenario,” explains Dr. Pendharkar.
The impact of this line of research is on the larger effort on the classification of such algebras. “It is my hope that as an Operator Algebraist and a new faculty member at the university, I bring this breadth to the department that has a very strong Mathematical Analysis research group,” he continues.
Pendharkar concludes that while this project has been in the in the works since 1999, that it’s “one more reason to feel accomplished as a human being in that, it’s a reminder that we can stay focused on a problem or advancement of a theory, whether or not we solve or advance it in our lifetime.”