We generalize the influential $C^*$-algebraic results of Kawamura–Tomiyama and Archbold–Spielberg for crossed products of discrete groups actions to the realm of Banach algebras and twisted actions. We prove that topological freeness is equivalent to the intersection property for all reduced twisted Banach algebra crossed products coming from subgroups, and in the untwisted case to a generalized intersection property for a full $L^p$-operator algebra crossed product for any $p\in [1,\,\infty ]$. This gives efficient simplicity criteria for various Banach algebra crossed products. We also use it to identify the prime ideal space of some crossed products as the quasi-orbit space of the action. For amenable actions we prove that the full and reduced twisted $L^p$-operator algebras coincide.

In this paper, we apply the theory of algebraic cohomology to study the amenability of Thompson’s group $\mathcal {F}$. We introduce the notion of unique factorization semigroup which contains Thompson’s semigroup $\mathcal {S}$ and the free semigroup $\mathcal {F}_n$ on n ($\geq 2$) generators. Let $\mathfrak {B}(\mathcal {S})$ and $\mathfrak {B}(\mathcal {F}_n)$ be the Banach algebras generated by the left regular representations of $\mathcal {S}$ and $\mathcal {F}_n$, respectively. We prove that all derivations on $\mathfrak {B}(\mathcal {S})$ and $\mathfrak {B}(\mathcal {F}_n)$ are automatically continuous, and every derivation on $\mathfrak {B}(\mathcal {S})$ is induced by a bounded linear operator in $\mathcal {L}(\mathcal {S})$, the weak-operator closed Banach algebra consisting of all bounded left convolution operators on $l^2(\mathcal {S})$. Moreover, we prove that the first continuous Hochschild cohomology group of $\mathfrak {B}(\mathcal {S})$ with coefficients in $\mathcal {L}(\mathcal {S})$ vanishes. These conclusions provide positive indications for the left amenability of Thompson’s semigroup.

We prove that a Banach algebra B that is a completion of the universal enveloping algebra of a finite-dimensional complex Lie algebra $\mathfrak {g}$ satisfies a polynomial identity if and only if the nilpotent radical $\mathfrak {n}$ of $\mathfrak {g}$ is associatively nilpotent in B. Furthermore, this holds if and only if a certain polynomial growth condition is satisfied on $\mathfrak {n}$ .

Non-amenability of ${\mathcal {B}}(E)$ has been surprisingly difficult to prove for the classical Banach spaces, but is now known for E = ℓp and E = Lp for all 1 ⩽ p < ∞. However, the arguments are rather indirect: the proof for L1 goes via non-amenability of $\ell ^\infty ({\mathcal {K}}(\ell _1))$ and a transference principle developed by Daws and Runde (Studia Math., 2010).

In this note, we provide a short proof that ${\mathcal {B}}(L_1)$ and some of its subalgebras are non-amenable, which completely bypasses all of this machinery. Our approach is based on classical properties of the ideal of representable operators on L1, and shows that ${\mathcal {B}}(L_1)$ is not even approximately amenable.

We initiate a study of structural properties of the quotient algebra ${\mathcal{K}}(X)/{\mathcal{A}}(X)$ of the compact-by-approximable operators on Banach spaces $X$ failing the approximation property. Our main results and examples include the following: (i) there is a linear isomorphic embedding from $c_{0}$ into ${\mathcal{K}}(Z)/{\mathcal{A}}(Z)$, where $Z$ belongs to the class of Banach spaces constructed by Willis that have the metric compact approximation property but fail the approximation property, (ii) there is a linear isomorphic embedding from a nonseparable space $c_{0}(\unicode[STIX]{x1D6E4})$ into ${\mathcal{K}}(Z_{FJ})/{\mathcal{A}}(Z_{FJ})$, where $Z_{FJ}$ is a universal compact factorisation space arising from the work of Johnson and Figiel.

Motivated by the definition of a semigroup compactication of a locally compact group and a large collection of examples, we introduce the notion of an (operator) homogeneous left dual Banach algebra (HLDBA) over a (completely contractive) Banach algebra $A$. We prove a Gelfand-type representation theorem showing that every HLDBA over A has a concrete realization as an (operator) homogeneous left Arens product algebra: the dual of a subspace of $A^{\ast }$ with a compatible (matrix) norm and a type of left Arens product $\Box$. Examples include all left Arens product algebras over $A$, but also, when $A$ is the group algebra of a locally compact group, the dual of its Fourier algebra. Beginning with any (completely) contractive (operator) $A$-module action $Q$ on a space $X$, we introduce the (operator) Fourier space $({\mathcal{F}}_{Q}(A^{\ast }),\Vert \cdot \Vert _{Q})$ and prove that $({\mathcal{F}}_{Q}(A^{\ast })^{\ast },\Box )$ is the unique (operator) HLDBA over $A$ for which there is a weak$^{\ast }$-continuous completely isometric representation as completely bounded operators on $X^{\ast }$ extending the dual module representation. Applying our theory to several examples of (completely contractive) Banach algebras $A$ and module operations, we provide new characterizations of familiar HLDBAs over A and we recover, and often extend, some (completely) isometric representation theorems concerning these HLDBAs.

The object of this paper is to prove a version of the Beurling–Helson–Lowdenslager invariant subspace theorem for operators on certain Banach spaces of functions on a multiply connected domain in $\mathbb{C}$. The norms for these spaces are either the usual Lebesgue and Hardy space norms or certain continuous gauge norms. In the Hardy space case the expected corollaries include the characterization of the cyclic vectors as the outer functions in this context, a demonstration that the set of analytic multiplication operators is maximal abelian and reflexive, and a determination of the closed operators that commute with all analytic multiplication operators.

A locally compact group G is compact if and only if its convolution algebras contain non-zero (weakly) completely continuous elements. Dually, G is discrete if its function algebras contain non-zero completely continuous elements. We prove non-commutative versions of these results in the case of locally compact quantum groups.

We show in a straightforward way that the non-commutative Schwartz space is weakly amenable. At the end, we leave an open problem.

We study when certain properties of Banach algebras are stable under ultrapower constructions. In particular, we consider when every ultrapower of is Arens regular, and give some evidence that this is so if and only if is isomorphic to a closed subalgebra of operators on a super-reflexive Banach space. We show that such ideas are closely related to whether one can sensibly define an ultrapower of a dual Banach algebraffi We study how tensor products of ultrapowers behave, and apply this to study the question of when every ultrapower of is amenable. We provide an abstract characterization in terms of something like an approximate diagonal, and consider when every ultrapower of a C*-algebra, or a group L1-convolution algebra, is amenable.

This note improves two previous results of the second author. They turn out to be special cases of our main theorem which states: A Banach space X has the property that the strong closure of every abstractly σ-complete Boolean algebra of projections in X is Bade complete if and only if X does not contain a copy of the sequence space ℓ∞.

Let B(X) denote the Banach algebra of all bounded linear operators on a Banach space X. We show that B(X) is finite if and only if no proper, complemented subspace of X is isomorphic to X, and we show that B(X) is properly infinite if and only if X contains a complemented subspace isomorphic to X[oplus ]X. We apply these characterizations to find Banach spaces X1, X2, and X3 such that B(X1) is finite, B(X2) is infinite, but not properly infinite, and B(X3) is properly infinite. Moreover, we prove that every unital, properly infinite ring has a continued bisection of the identity, and we give examples of Banach spaces D1 and D2 such that B(D1) and B(D2) are infinite without being properly infinite, B(D1) has a continued bisection of the identity, and B(D2) has no continued bisection of the identity. Finally, we exhibit a unital $C^\ast$-algebra which is finite and has a continued bisection of the identity.