We investigate strongly continuous one-parameter (C0) groups of isometries acting on certain spaces of analytical functions which were introduced by Kolaski (C. J. Kolaski, Isometries of some smooth normed spaces of analytic functions, Complex Var. Theory Appl. 10(2–3) (1988), 115–122). We characterize the generators of these groups of isometries and also the spectrum of the generators. We provide an example on the Bloch space of an unbounded hermitian operator with non-compact resolvent.

In the sequel we construct simple, unital, separable, stable, amenable ${{C}^{*}}$-algebras for which the ordered ${{K}_{0}}$-group is strongly perforated and group isomorphic to $Z$. The particular order structures to be constructed will be described in detail below, and all known results of this type will be generalised.

A Banach space satisfying some physically significant geometric properties is shown to be the predual of a JBW*–triple. If one considers the unit ball of this Banach space as the state space of a physical system, the result shows that the set of observables is equipped with a natural ternary algebraic structure. This provides a spectral theory and other tools for studying the quantum mechanical measuring process

A norm |⋅| on a Banach space X is locally uniformly rotund (LUR) if lim |xn — x| = 0 for every xn, x ∈ X for which lim2|x|2 + 2 |xn|2-|xn+xn|2 = 0. It is shown that a Banach space X admits an equivalent LUR norm provided there is a bounded linear operator T of X into c0(Γ) such that T* c*0(Γ) is norm dense in X*. This is the case e.g. if X* is weakly compactly generated (WCG).