We study disjointness preserving (quasi-) n-shift operators on C 0 (X), where X is locally compact and Hausdorff. When C 0 (X) admits a quasi-n-shift T, there is a countable subset of X= X{} equipped with a tree-like structure, called -tree, with exactly n joints such that the action of T on C 0 (X) can be implemented as a shift on the -tree. If T is an n-shift, then the -tree is dense in X and thus X is separable. By analyzing the structure of the -tree, we show that every (quasi-) n-shift on c 0 can always be written as a product of n (quasi-) 1-shifts. Although it is not the case for general C 0 (X) as shown by our counter examples, we can do so after dilation.

Inspired by the recent work of Bekka, we study two reasonable analogues of property (T) for not necessarily unital C-algebras. The stronger one of the two is called property (T) and the weaker one is called property (T e). It is shown that all non-unital C-algebras do not have property (T)(neither do their unitalizations). Moreover, all non-unital -unital C-algebras do not have property (T e).

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Let E be a real Banach space with a uniformly Gteaux differentiable norm and which possesses uniform normal structure, K a nonempty bounded closed convex subset of E,{T i} i= 1 N a finite family of asymptotically nonexpansive self-mappings on K with common sequence {k n} n= 1[1,),{t n},{s n} be two sequences in (0, 1) such that s n+ t n= 1 (n 1) and f be a contraction on K. Under suitable conditions on the sequences {s n},{t n}, we show the existence of a sequence {x n} satisfying the relation x n=(1 1 k n) x n+ s n k n f (x n)+ t n k n T r n n x n where n= l n N+ r n for some unique integers l n 0 and 1 r n N. Further we prove that {x n} converges strongly to a common fixed point of {T i} i= 1 N, which solves some variational inequality, provided x n T i x n 0 as n for i= 1, 2,, N. As an application, we prove that the iterative process defined by z 0 K, z n+ 1=(1 1 k n) z n+ s n k n f (z n)+ t n k n

The notion of generating topologies, introduced by Stephani El0], is useful for studying the injective hull of an operator ideal. Using Randtke's idea (see [8, p. 90] or [12,(3.2. 1) and (3.2. 7)]), we can characterize generating topologies in terms of seminorms which satisfy some expected properties (see Lemma 3.3 and Theorems 3.4 and 3.9). By a well-known and useful idea of Grothendieck, the dual notions of generating topologies and ideal-topologies, the so-called generating bornologies, are given and studied in Sect. 4. In term of ideal-bornologies, the surjective hull of an operator ideal on Banach spaces is given (see Lemma 4.8 and Theorem 4.10). In terms of these two dual concepts, we are able to classify locally convex spaces, and to study their dual results. For instance, we show that if 9.1 is a symmetric (resp. completely symmetric) operator ideal on Banach spaces then a Banach space E is an 9~-topological