The Peter–Weyl idempotent $e_{\mathscr{P}}$ of a parahoric subgroup $\mathscr{P}$ is the sum of the idempotents of irreducible representations of $\mathscr{P}$ that have a nonzero Iwahori fixed vector. The convolution algebra associated with $e_{\mathscr{P}}$ is called a Peter–Weyl Iwahori algebra. We show that any Peter–Weyl Iwahori algebra is Morita equivalent to the Iwahori–Hecke algebra. Both the Iwahori–Hecke algebra and a Peter–Weyl Iwahori algebra have a natural conjugate linear anti-involution $\star$, and the Morita equivalence preserves irreducible hermitian and unitary modules. Both algebras have another anti-involution, denoted by $\bullet$, and the Morita equivalence preserves irreducible and unitary modules for $\bullet$.

In this paper we introduce a new way of deforming convolution algebras and Fourier algebras on locally compact groups. We demonstrate that this new deformation allows us to reveal some information about the underlying groups by examining Banach algebra properties of deformed algebras. More precisely, we focus on representability as an operator algebra of deformed convolution algebras on compact connected Lie groups with connection to the real dimension of the underlying group. Similarly, we investigate complete representability as an operator algebra of deformed Fourier algebras on some finitely generated discrete groups with connection to the growth rate of the group.

There is an error in one of the major results in our original paper ‘Equivalent weights and standard homomorphisms for convolution algebras on ℝ+’. We describe the error and give a counterexample to the result as stated. We then give a substitute result which is in many ways stronger than the erroneous result. We will also indicate what changes need to be made in the original paper to accommodate the replacement of the erroneous result by the substitute.

We take a second look at two basic topics in the study of weighted convolution algebras L1(ω) on ℝ+. An early result showed that one could replace the weight $\omega$ with a very well-behaved weight without changing the space L1(ω) as long as L1(ω) was an algebraffi We prove the analogous result for measure algebras when M(ω) is an algebraffi This allows us to preserve not only the norm topology but also the relative weak* topology on L1(ω). A homomorphism between weighted convolution algebras is said to be standard if it preserves generators of dense principal ideals. The original proofs of standardness and its variants are all based on finding the generator of a particular strongly continuous convolution semigroup. In this paper we give much simpler direct proofs of these results. We also improve the statement and proof of the theorem, giving useful properties equivalent to the standardness of a homomorphism.

In this paper we derive precise tail-area approximations for the sum of an arbitrary finite number of independent heavy-tailed random variables. In order to achieve second-order asymptotics, a mild regularity condition is imposed on the class of distribution functions with regularly varying tails.

Higher-order asymptotics are also obtained when considering asemiparametric subclass of distribution functions with regularly varying tails. These semiparametric subclasses are shown to be closed under convolutions and a convolution algebra is constructed to evaluate the parameters of a convolution from the parameters of the constituent distributions in the convolution. A Maple code is presented which does this task.