We present two examples. One of an operator T such that ${T^n(T-I)}_{n=1}^{\infty }$ is precompact in the operator norm and the spectrum of T on the unit circle consists of an infinite number of points accumulating at 1, and the other of an operator T such that ${T^n(T-I)}_{n=1}^{\infty }$ is convergent to zero but T is not power bounded.

We compare the singular spectrum of a Banach algebra element with the usual spectrum and exponential spectrum.

A Banach algebra A is said to be topologically nilpotent if $sup\parallel x₁......x_n{\parallel }^{1/n}:x_i\in A,\parallel x_i\parallel \le 1\left(1\le i\le n\right)$ tends to 0 as n → ∞. We continue the study of topologically nilpotent algebras which was started in [2]

A simple and natural example is given of a non-commuting Arens multiplication.

Denote by any set of cardinality continuum. It is proved that a Banach algebra A with the property that for every collection $a_{\alpha }:\alpha \in \subset A$ there exist α ≠ β ∈ such that $a_{\alpha }\in a_{\beta }{A}^{}$ is isomorphic to ${⨁}_{i=1}^r(ℂ\left[X\right]/X^{d_i}ℂ\left[X\right])\oplus E$, where $d₁,...,d_r\in \mathbb{N}$, and E is either $Xℂ\left[X\right]/X^{d₀}ℂ\left[X\right]$ for some d₀ ∈ ℕ or a 1-dimensional ${⨁}_{i=1}^rℂ\left[X\right]/X^{d_i}ℂ\left[X\right]$-bimodule with trivial right module action. In particular, ℂ is the unique non-zero prime Banach algebra satisfying the above condition.

In this paper weare interested in subsets of a real Banach space on which different classes of functions are bounded.

Let A be a locally convex, unital topological algebra whose group of units $A^{\times }$ is open and such that inversion $\iota :A^{\times }\to A^{\times }$ is continuous. Then inversion is analytic, and thus $A^{\times }$ is an analytic Lie group. We show that if A is sequentially complete (or, more generally, Mackey complete), then $A^{\times }$ has a locally diffeomorphic exponential function and multiplication is given locally by the Baker-Campbell-Hausdorff series. In contrast, for suitable non-Mackey complete A, the unit group $A^{\times }$ is an analytic Lie group without...