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Suppose that $H$ is a complex Hilbert space and that $\mathcal{B}(H)$ denotes the bounded linear operators on $H$ . We show that every abelian, amenable operator algebra is similar to a $C^{*}$ -algebra. We do this by showing that if $\mathcal{A}\subseteq\mathcal{B}(H)$ is an abelian algebra with the property that given any bounded representation $\varrho:\mathcal{A}\to\mathcal{B}(H_{\varrho})$ of $\mathcal{A}$ on a Hilbert space $H_{\varrho}$ , every invariant subspace of $\varrho(\mathcal{A})$ is topologically complemented by another invariant subspace of $\varrho(\mathcal{A})$ , then $\mathcal{A}$ is similar to an abelian $C^{*}$ -algebra.

This paper considers Banach algebras with properties 𝔸 or 𝔹, introduced recently by Alaminos et al. The class of Banach algebras satisfying either of these two properties is quite large; in particular, it includes C*-algebras and group algebras on locally compact groups. Our first main result states that a continuous orthogonally additive n-homogeneous polynomial on a commutative Banach algebra with property 𝔸 and having a bounded approximate identity is of a standard form. The other main results describe Banach algebras A with property 𝔹 and having a bounded approximate identity that admit non-zero continuous symmetric orthosymmetric n-linear maps from An into ℂ.

Let $A$ be a complex Banach algebra. If the spectrum of an invertible element $a\in A$ does not separate the plane, then $a$ admits a logarithm. We present two elementary proofs of this classical result which are independent of the holomorphic functional calculus. We also discuss the case of real Banach algebras. As applications, we obtain simple proofs that every invertible matrix over $\mathbb C$ has a logarithm and that every real matrix $M$ in $M_n(\mathbb R)$ with $\det M>0$ is a product of two real exponential matrices., 9 pages

For semisimple, commutative Banach algebras $\mathcal A$ and $\mathcal B$, some Banach algebra properties of the Cartesin product $\mathcal A \times \mathcal B$ are characterized in terms of $\mathcal A$ and $\mathcal B$. A couple of results are also proved for non-commutative Banach algebras.

We consider pseudoquotient extensions of positive linear functionals on a commutative Banach algebra $\mathcal{A}$ and give conditions under which the constructed space of pseudoquotients can be identified with all Radon measures on the structure space $\hat{\mathcal{A}}$.

Every element 6 of a radical Banach algebra & satisfies l iπv^ ||6||=0. We are concerned here with the existence of a lower bound for the rate of decrease of the sequence (||δ||) under various assumptions over b and ^ , when b is not nilpotent and & commutative. If the nilpotents are dense in & then for every sequence (λn) of positive reals there exists a nonnilpotent b e & such that lim inf*-«, ||6||/ϋ»=0. A stronger result holds if & possesses furthermore a bounded approximate identity. On the other direction if & has no nilpotent element and if some element of & which is not a divisor of zero acts compactly on & then there exists a sequence (2n) of positive reals such that \immfn-+0O\\b n\\lλn— + °o for every nonzero be&. Also there exists universal lower bounds for the rate of decrease of ll