# C* Algebra

### From Mathsreach

**Download article: Examples from C* Algebra**(IMAges Issue 11: October 2011)

**Professor Astrid an Huef works in a relatively young branch of mathematics.**

**nounced C star) is a branch of mathematics concerning special kinds of operator algebras, which derived from attempts in the 1930s to understand the new and startling challenges of quantum mechanics. C*- algebra uses the tools of functional analysis to study problems associated with infinite-dimensional analogues of linear algebra, where the operators are linear transformations with extra properties.**

Since the 1950s, operator algebras have provided a fundamental tool in areas such as representation theory, Fourier analysis and dynamics. More recently their use has led to surprising successes in other parts of mathematics, including algebraic topology, number theory, and combinatorics.

“Many of the operator algebras that I am interested in arise from dynamical systems - mathematical objects that model the way things change,” says an Huef. “The challenge is to deduce information about the algebra from the dynamical system and vice versa. I’m interested in systems like transformation groups, groupoids and higher-rank graphs.”

“There are very beautiful structure theories which relate properties of these dynamical systems to those of the associated algebras. Because we prove theorems that set up a one-to-one correspondence between properties of the system and properties of the algebra, I can tell colleagues what sort of dynamical system they need to start with to get the properties they want.”

Most of an Huef ’s research has been about reversible dynamical systems, in which time can go backwards and forwards. More recently she has studied irreversible systems, which only go forwards. “We kept on finding interesting examples where the techniques developed for studying reversible systems were ineffective, and eventually decided to develop a general theory. We believe that this will become an increasingly important aspect of the subject. And it is easy to point to important irreversible processes: everything ages, and many of the effects are irreversible.”

When an Huef arrived at the University of Otago from Australia almost two years ago, there was no one else working in this area of mathematics in New Zealand, and she was collaborating with colleagues from Australia, the United States, Scotland and Brazil. She is now working with an algebraist in her new department “in a purely algebraic setting, which is new to me. New Zealand has some world-class algebraists, and I am looking forward to learning more about pure algebra.” She has already built up a lively research group at Otago, and their working seminar is already producing interesting new results.

An Huef was born in Germany but “always had fond memories” of her two years as a teenager living in Wellington and going to Wellington High School. She has one female collaborator and says the number of women in pure mathematics is increasing. “I used to be the only one at some conferences, but now you need to queue for the women’s bathroom.”