Maliheh Hosseini

On isometries of absolutely continuous function algebras in compact and non-compact frameworks.

Let X be a subset of the real line \Bbb R with at least two points. A complex-valued function f on X is of bounded variation if the total variation \mathcal{V}(f) of f is finite, i.e.,

\left{ \mathcal{V}(f):=\sup \left\{\displaystyle\sum_{i=1}^{n} |f(x_i)-f(x_{i-1})|: n\in \Bbb N, x_0,\dots, x_n\in X,x_0 < \dots < x_n \right\} < \infty \right}

Moreover, a function f:X\to \Bbb C is called absolutely continuous if given \epsilon>0, there exists a \delta>0 such that \sum_{i=1}^{n} |f(b_i)-f(a_{i})|<\epsilon, for every finite family of non-overlapping open intervals \{(a_i,b_i):i=1,\cdots,n\} whose extreme points belong to X with \sum_{i=1}^{n} (b_i-a_i)<\delta. We denote by AC_b(X) the algebra of all absolutely continuous functions of bounded variation on X. Note that when X is bounded, each absolutely continuous function is automatically of bounded variation. For the case where X is compact, we write AC(X) instead of AC_b(X).

In this talk I review the main results concerning surjective linear isometries of AC_b(X)-algebras equipped with two natural norms, the sum-norm \|\cdot\|_\infty+\mathcal{V}(\cdot) and the max-norm \max\{\|\cdot\|_\infty,\mathcal{V}(\cdot)\}, where \|\cdot\|_\infty denotes the supremum norm. Actually, with the class of isometries we make a connection between the algebraic and the topological structures of AC_b(X)-algebras.