We introduce and study the weak Glimm property for $\mathrm{C}^{*}$-algebras, and also a property we shall call (HS$_0$). We show that the properties of being nowhere scattered and residual (HS$_0$) are equivalent for any $\mathrm{C}^{*}$-algebra. Also, for a $\mathrm{C}^{*}$-algebra with the weak Glimm property, the properties of being purely infinite and weakly purely infinite are equivalent. It follows that for a $\mathrm{C}^{*}$-algebra with the weak Glimm property such that the absolute value of every nonzero, square-zero, element is properly infinite, the properties of being (weakly, locally) purely infinite, nowhere scattered, residual (HS$_0$), residual (HS$_{\text {t}}$), and residual (HI) are all equivalent, and are equivalent to the global Glimm property. This gives a partial affirmative answer to the global Glimm problem, as well as certain open questions raised by Kirchberg and Rørdam.

Let G be a group that is either virtually soluble or virtually free, and let ω be a weight on G. We prove that if G is infinite, then there is some maximal left ideal of finite codimension in the Beurling algebra $\ell^1(G, \omega)$, which fails to be (algebraically) finitely generated. This implies that a conjecture of Dales and Żelazko holds for these Banach algebras. We then go on to give examples of weighted groups for which this property fails in a strong way. For instance, we describe a Beurling algebra on an infinite group in which every closed left ideal of finite codimension is finitely generated and which has many such ideals in the sense of being residually finite dimensional. These examples seem to be hard cases for proving Dales and Żelazko’s conjecture.

By defining and applying the restricted topology, we have investigated certain connections between the boundary spectrum, the exponential spectrum, the topological boundary of the spectrum and the connected hull of the spectrum (see Mouton and Harte [‘Linking the boundary and exponential spectra via the restricted topology’, J. Math. Anal. Appl. 454 (2017), 730–745]). We now solve a remaining problem regarding the restricted connected hull.

Let ${\mathcal A}$ be a Banach algebra, and let $\varphi $ be a nonzero character on ${\mathcal A}$. For a closed ideal I of ${\mathcal A}$ with $I\not \subseteq \ker \varphi $ such that I has a bounded approximate identity, we show that $\operatorname {WAP}(\mathcal {A})$, the space of weakly almost periodic functionals on ${\mathcal A}$, admits a right (left) invariant $\varphi $-mean if and only if $\operatorname {WAP}(I)$ admits a right (left) invariant $\varphi |_I$-mean. This generalizes a result due to Neufang for the group algebra $L^1(G)$ as an ideal in the measure algebra $M(G)$, for a locally compact group G. Then we apply this result to the quantum group algebra $L^1({\mathbb G})$ of a locally compact quantum group ${\mathbb G}$. Finally, we study the existence of left and right invariant $1$-means on $ \operatorname {WAP}(\mathcal {T}_{\triangleright }({\mathbb G}))$.

We characterise the existence of certain (weakly) compact multipliers of the second dual of symmetric abstract Segal algebras in both the group algebra $L^{1}(G)$ and the Fourier algebra $A(G)$ of a locally compact group G.

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 add to the list of Banach spaces X for which it is known that the space of bounded linear operators on X has a unique maximal ideal. In particular, the result holds if X is a subsymmetric direct sum of ℓp or of the Schlumprecht space S. We also show that two recently identified ideals in L(Jp), where Jp is the pth James space, each contains a unique maximal ideal.

In this paper we first show that for a locally compact amenable group $G$, every proper abstract Segal algebra of the Fourier algebra on $G$ is not approximately amenable; consequently, every proper Segal algebra on a locally compact abelian group is not approximately amenable. Then using the hypergroup generated by the dual of a compact group, it is shown that all proper Segal algebras of a class of compact groups including the $2\times 2$ special unitary group, $\mathrm{SU} (2)$, are not approximately amenable.

For ${{C}^{*}}$-algebras $\mathcal{A}$ of real rank zero, we describe linear maps $\phi $ on $\mathcal{A}$ that are surjective up to ideals $\mathcal{J}$, and $\text{ }\pi \text{ (}A\text{)}$ is invertible in $\mathcal{A}/\mathcal{J}$ if and only if $\text{ }\pi \text{ (}\phi (A))$ is invertible in $\mathcal{A}/\mathcal{J}$, where $A\,\in \,\mathcal{A}$ and $\text{ }\pi \text{ }\text{:}\mathcal{A}\to \mathcal{A}\text{/}\mathcal{J}$ is the quotient map. We also consider similar linear maps preserving zero products on the Calkin algebra.

We investigate generalized amenability and biflatness properties of various (operator) Segal algebras in both the group algebra, ${{L}^{1}}\left( G \right)$, and the Fourier algebra, $A\left( G \right)$, of a locally compact group $G$.

In this paper we begin with a short, direct proof that the Banach algebra B(l1) is not amenable. We continue by showing that various direct sums of matrix algebras are not amenable either, for example the direct sum of the finite dimensional algebras is no amenable for 1 ≤ p ≤ ∞, p ≠ 2. Our method of proof naturally involves free group algebras, (by which we mean certain subalgebras of B(X) for some space X with symmetric basis—not necessarily X = l2) and we introduce the notion of ‘relative amenability’ of these algebras.

Let $\idemp$ be the set of idempotents in a finite-dimensional real algebra $A$. Let $p$ and $q$ be idempotents that lie in the same component of $\idemp$. Then, among the continuous paths connecting $p$ and $q$ in $\idemp$, there exist a polynomial path of degree at most $3$ and a polygonal path consisting of at most three segments.

A well-known theorem of Sarason [11] asserts that if $\left[ {{T}_{f}},\,{{T}_{h}} \right]$ is compact for every $h\,\in \,{{H}^{\infty }}$, then $f\,\in \,{{H}^{\infty }}\,+\,C\left( T \right)$. Using local analysis in the full Toeplitz algebra $T\,=\,T\left( {{L}^{\infty }} \right)$, we show that the membership $f\,\in \,{{H}^{\infty }}\,+\,C\left( T \right)$ can be inferred from the compactness of a much smaller collection of commutators $\left[ {{T}_{f}},\,{{T}_{h}} \right]$. Using this strengthened result and a theorem of Davidson [2], we construct a proper ${{C}^{*}}$-subalgebra $T\left( \mathcal{L} \right)$ of $T$ which has the same essential commutant as that of $T$. Thus the image of $T\left( \mathcal{L} \right)$ in the Calkin algebra does not satisfy the double commutant relation [12], [1]. We will also show that no separable subalgebra $\mathcal{S}$ of $T$ is capable of conferring the membership $f\,\in \,{{H}^{\infty }}\,+\,C\left( T \right)$ through the compactness of the commutators $\left\{ \left[ {{T}_{f,}}\,S \right]\,:\,S\,\in \,\mathcal{S} \right\}$.

We show that, if a Banach algebra $\mathfrak{A}$ is a left ideal in its second dual algebra and has a left bounded approximate identity, then the weak amenability of $\mathfrak{A}$ implies the $\left( 2m+1 \right)$-weak amenability of $\mathfrak{A}$ for all $m\,\ge \,1$.