We state best approximation and fixed point theorems in modular spaces endowed with an H-space structure given by the modular topology. We consider both the cases of single valued functions and multifunctions. These theorems extend some previous results due to Ky Fan.

The second dual of the vector-valued function space C0(S, A) is characterized in terms of generalized functions in the case where A* and A** have the Radon-Nikodým property. As an application we present a simple proof that C0 (S, A) is Arens regular if and only if A is Arens regular in this case. A representation theorem of the measure μh is given, where , h ∈ L∞ (|μ;|, A**) and μh is defined by the Arens product.

An example of two disjoint special classes whose upper radicals coincide is presented. It is shown that the left hereditary subradical of the hereditary idempotent radical is right hereditary. An example of a hereditary and principally left hereditary radical which is not left hereditary is constructed.

We consider coactions of a locally compact group G on a C*-algebra A, and the associated crossed product C*-algebra A× G. Given a normal subgroup N of G, we seek to decompose A× G as an iterated crossed product (A× G/ N) × N, and introduce notions of twisted coaction and twisted crossed product which make this possible. We then prove a duality theorem for these twisted crossed products, and discuss how our results might be used, especially when N is abelian.

Let Δ be a thick building of type Ã2, and let be its set of vertices. We study a commutative algebra of ‘averaging’ operators acting on the space of complex valued functions on . This algebra may be identified with a space of ‘biradial functions’ on , or with a convolution algebra of bi-K-invariant functions on G, if G is a sufficiently large group of ‘type-rotating’ automorphisms of Δ, and K is the subgroup of G fixing a given vertex. We describe the multiplicative functionals on and the corresponding spherical functions. We consider the C*-algebra induced by on l2, find its spectrum Σ, prove positive definiteness of a kernel kz for each z ∈ Σ, find explicity the spherical Plancherel formula for any group G of type rotating automorphisms, and discuss the irreducibility of the unitary representations appearing therein. For the class of buildings ΔJ arising from the groups ΓJ introduced in [2], this involves proving that the weak closure of is maximal abelian in the von Neumann algebra generated by the left regular representation of ΓJ.

It is proved that on certain kinds of homogeneous spaces, the only Lp function, 1≤ p < ∞, satisfying the mean value property is the zero function.

An infinite family of vertex-and edge-transitive, but not arc-transitive, graphs of degree 4 is constructed.

Using Fourier transforms, we give a new proof of certain identites for the fundamental solutions of the iterated Dirac operators and Ḏl = (Σ/Σx0 + Ḏ)l. Based on the close relationship between the fundamental solutions and the conformal weights we then give a simple proof of B. Bojarski's results on the conformal covariance of Ḏl. We also prove a new conformal covatiance result of D.

If h is the composition of two formal power series f and g, and if h is even, what can be said about f and g? Some partial answers are given here.