Noncommutative Analysis

On the isomorphism question for complete Pick multiplier algebras

In this post I want to tell you about our new preprint, “On the isomorphism question for complete Pick multiplier algebras“,  which Matt Kerr, John McCarthy and myself just uploaded to the arXiv. Very broadly speaking, the motif of this paper is the connection between algebra and geometry; to be a little bit more precise, it is the connection between complex geometry and nonself-adjoint operator algebras. Read the rest of this entry »

Advanced Analysis, Notes 8: Banach spaces (application: weak solutions to PDEs)

Today I will show you an application of the Hahn-Banach Theorem to partial differential equations (PDEs). I learned this application in a seminar in functional analysis, run by Haim Brezis, that I was fortunate to attend in the spring of 2008 at the Technion.

As often happens with serious applications of functional analysis, there is some preparatory material to go over, namely, weak solutions to PDEs.

Read the rest of this entry »

Advanced Analysis, Notes 7: Banach spaces (dual spaces and duality, Lp spaces, the double dual, quotient spaces)

Today we continue our treatment of the dual space X^* of a normed space (usually Banach) X. We start by considering a wide class of Banach spaces and their duals.  Read the rest of this entry »