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.

Let $\mathbb{G}$ be a locally compact quantum group and let $I$ be a closed ideal of $L^{1}(\mathbb{G})$ with $y|_{I}\neq 0$ for some $y\in \text{sp}(L^{1}(\mathbb{G}))$. In this paper, we give a characterization for compactness of $\mathbb{G}$ in terms of the existence of a weakly compact left or right multiplier $T$ on $I$ with $T(f)(y|_{I})\neq 0$ for some $f\in I$. Using this, we prove that $I$ is an ideal in its second dual if and only if $\mathbb{G}$ is compact. We also study Arens regularity of $I$ whenever it has a bounded left approximate identity. Finally, we obtain some characterizations for amenability of $\mathbb{G}$ in terms of the existence of some $I$-module homomorphisms on $I^{\ast \ast }$ and on $I^{\ast }$.

We consider the unital Banach algebra $\ell ^{1}(\mathbb{Z}_{+})$ and prove directly, without using cyclic cohomology, that the simplicial cohomology groups ${\mathcal{H}}^{n}(\ell ^{1}(\mathbb{Z}_{+}),\ell ^{1}(\mathbb{Z}_{+})^{\ast })$ vanish for all $n\geqslant 2$. This proceeds via the introduction of an explicit bounded linear operator which produces a contracting homotopy for $n\geqslant 2$. This construction is generalised to unital Banach algebras $\ell ^{1}({\mathcal{S}})$, where ${\mathcal{S}}={\mathcal{G}}\cap \mathbb{R}_{+}$ and ${\mathcal{G}}$ is a subgroup of $\mathbb{R}_{+}$.

We characterize two important notions of amenability and compactness of a locally compact quantum group $\mathbb{G}$ in terms of certain homological properties. For this, we show that $\mathbb{G}$ is character amenable if and only if it is both amenable and co-amenable. We finally apply our results to Arens regularity problems of the quantum group algebra ${{L}^{1}}\left( \mathbb{G} \right)$. In particular, we improve an interesting result by Hu, Neufang, and Ruan.

We prove that no proper Segal algebra of a SIN group is approximately amenable.

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 $G$.

Let $\mathcal{H}$ be a complex separable Hilbert space and $\mathcal{L}\left( \mathcal{H} \right)$ denote the collection of bounded linear operators on $\mathcal{H}$. In this paper, we show that for any operator $A\,\in \,\mathcal{L}\left( \mathcal{H} \right)$, there exists a stably finitely $\left( \text{SI} \right)$ decomposable operator ${{A}_{\epsilon }}$, such that $\left\| A-{{A}_{\epsilon }} \right\|\,<\,\epsilon$ and ${{\mathcal{A}}^{\prime }}\left( {{A}_{\epsilon }} \right)/\text{rad}\,{{\mathcal{A}}^{\prime }}\left( {{A}_{\epsilon }} \right)$ is commutative, where rad ${{\mathcal{A}}^{\prime }}\left( {{A}_{\epsilon }} \right)$ is the Jacobson radical of ${{\mathcal{A}}^{\prime }}\left( {{A}_{\epsilon }} \right)$. Moreover, we give a similarity classification of the stably finitely decomposable operators that generalizes the result on similarity classification of Cowen–Douglas operators given by C. L. Jiang.

In this paper, we give a simple criterion for the Arens regularity of a bilinear mapping on normed spaces, which applies in particular to Banach module actions, and then investigate those conditions under which the second adjoint of a derivation into a dual Banach module is again a derivation. As a consequence of the main result, a simple and direct proof for several older results is also included.

We prove a quantized version of a theorem by M. V. Sheĭnberg: A uniform algebra equipped with its canonical, i.e., minimal, operator space structure is operator amenable if and only if it is a commutative ${{C}^{*}}$-algebra.

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 $m\,\ge \,1$.

A strongly maximal triangular AF algebra which is defined by a realvalued cocycle is said to be analytic. Formulas for generic cocycles are given separately for both the integer-valued case and the real-valued coboundary case, and also for certain nest algebras. In the case of an integer-valued cocycle, there is an associated partial homeomorphismof the maximal ideal space of the diagonal. If the partial homeomorphism extends to a homeomorphism, then the algebra embeds in a crossed product. This occurs for a large class of subalgebras of UHF algebras, but an example shows that this does not always occur. An example is given of a triangular AF algebra which is analytic via a coboundary but is not a nest algebra; also, it is shown that a nest algebra need not be analytic

Let X be a Banach space and let A be a uniformly closed algebra of compact operators on X, containing the finite rank operators. We set up a general framework to discuss the equivalence between Banach space approximation properties and the existence of right approximate identities in A. The appropriate properties require approximation in the dual X* by operators which are adjoints of operators on X. We show that the existence of a bounded right approximate identity implies that of a bounded left approximate identity. We give examples to show that these properties are not equivalent, however. Finally, we discuss the well known result that, if X* has a basis, then X has a shrinking basis. We make some attempts to generalize this to various bounded approximation properties.