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).

Properties of the subset of polygonal paths in the Hilbert space of paths referring to a<i>d</i>-dimensional quantum-mechanical system are examined. The results are used to discuss various types of polygonal-path approximations appearing in the functional-integration theory. The uniform approximation is applied to extend the definition of the Feynman maps from our previous paper and to prove consistency of this extension. Relations of the extended<i>F</i> _<i>i</i> map to the Wiener integral are given.

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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