Weak module amenability for the second dual of a Banach algebra

In this paper we study the weak module amenability of Banach algebras which are Banach modules over another Banach algebra with compatible actions. We show that for every module derivation D : A ↦ ( A/J_A )∗ if D∗∗(A∗∗) ⊆ WAP (A/J_A ), then weak module amenability of A∗∗ implies that of A. Also we prove that under certain conditions for the module derivation D, if A∗∗ is weak module amenable then A is also weak module amenable.

On measure algebras associated to locally compact groups

We shall consider measure algebras associated to locally compact groups, bounded operators between them and properties of the underlying measures. We take into account the second dual of measure algebras provided with the Arens products together with tools of Gélfand theory.

On the iterated normed duals

Several properties of the normed Hom-functor (dual) D and its iterations Dn are exhibited. For instance, D turns every canonical embedding (into the second dual space) to a retraction (of the third dual onto the first one) having for the right inverse the appropriate canonical embedding (of the first dual space into the third one). Some consequences to the direct-sum presentations and quotients of higher dual spaces are considered.

Evaluation of Standing Balance Performance in Indian Classical Dancers

Indian classical dance involves a constant change of the base of support from stance to low jumps and spins along with intricate footwork. Graceful movement of the torso, shifting from side to side and turning around the axis of the spine, challenges balance. Yet, balance performance remains unexplored in Indian classical dancers. Therefore, the present study aimed to compare the standing balance of 36 active female dancers (18 to 25 years of age) who had performed Indian classical dance for a minimum of 10 years with 36 healthy age-matched women not involved in regular physical activity. Balance was evaluated in static and dynamic conditions of single and dual-limb stance on a force plate using center-of-pressure trajectory and the Star Excursion Balance Test (SEBT). Dancers demonstrated better balance on both instrumented and non-instrumented outcome variables: wide base of support with eyes open and with eyes closed; for 30-second single limb stance with eyes open and with eyes closed; for 13-second dual task in single limb stance; and for 22-second dual task in wide base of support. The SEBT revealed significantly better balance performance of dancers in the three directions tested: anterior, posteromedial, and posterolateral. There was also a strength component of the study on which the dancers achieved significantly higher scores than controls for the three muscle groups tested (gastrocsoleus, gluteus medius, and quadriceps), which can be attributed to their training. These findings can be used to recommend classical dance training to achieve the dual purpose of deriving better balance and stronger bodies and maintaining the Indian dance heritage.

Ideals of the Quantum Group Algebra, Arens Regularity and Weakly Compact Multipliers

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 $I^{\ast }$.

ARENS PRODUCTS, ARENS REGULARITY AND RELATED PROBLEMS

We study the second dual algebra of a Banach algebra and related problems. We resolve some questions raised by Ülger, which are related to Arens products. We then discuss a question of Gulick on the radical of the second dual algebra of the group algebra of a discrete abelian group and give an application of Arens regularity to Fourier and Fourier–Stieltjes transforms.

Johnson pseudo-contractibility of second duals of certain Banach algebras

In this paper, we study Johnson pseudo-contractibility of second dual of some Banach algebras. We show that the semigroup algebra [Formula: see text] is Johnson pseudo-contractible if and only if [Formula: see text] is a finite amenable group, where [Formula: see text] is an archimedean semigroup. We also show that the matrix algebra [Formula: see text] is Johnson pseudo-contractible if and only if [Formula: see text] is finite. We study Johnson pseudo-contractibility of certain projective tensor product second duals Banach algebras.

A CHARACTERISATION OF TRACIALLY NUCLEAR C*-ALGEBRAS

We give two characterisations of tracially nuclear C*-algebras. The first is that the finite summand of the second dual is hyperfinite. The second is in terms of a variant of the weak* uniqueness property. The necessary condition holds for all tracially nuclear C*-algebras. When the algebra is separable, we prove the sufficiency.