Analogues of the classical Banach-Stone theorem for spaces of continuous functions are studied in the context of the spaces of absolutely continuous functions introduced by Ashton and Doust. We show that if AC(σ₁) is algebra isomorphic to AC(σ₂) then σ₁ is homeomorphic to σ₂. The converse however is false. In a positive direction we show that the converse implication does hold if the sets σ₁ and σ₂ are confined to a restricted collection of compact sets, such as the set of all simple polygons.

Suppose ℒ₁ and ℒ₂ are subspace lattices on complex separable Banach spaces X and Y, respectively. We prove that under certain lattice-theoretic conditions every isomorphism from algℒ₁ to algℒ₂ is quasi-spatial; in particular, if a subspace lattice ℒ of a complex separable Banach space X contains a sequence $E_i$ such that $\left(E_i\right)₋\ne X$, $E_i\subseteq E_{i+1}$, and ${\bigvee }_{i=1}^{\infty }{E}_i=X$ then every automorphism of algℒ is quasi-spatial.

We give a full description of the semiclassical spectral theory of quantum toric integrable systems using microlocal analysis for Toeplitz operators. This allows us to settle affirmatively the isospectral problem for quantum toric integrable systems: the semiclassical joint spectrum of the system, given by a sequence of commuting Toeplitz operators on a sequence of Hilbert spaces, determines the classical integrable system given by the symplectic manifold and commuting Hamiltonians. This type of...

Let $\mu$ be a measure on a domain $\Omega$ in ${ℂ}^n$ such that the Bergman space of holomorphic functions in $L^2(\Omega ,\mu )$ possesses a reproducing kernel $K(x,y)$ and $K(x,x)\>0$$\forall x\in \Omega$. The Berezin transform associated to $\mu$ is the integral...

We will discuss Kellogg's iterations in eigenvalue problems for normal operators. A certain generalisation of the convergence theorem is shown.