By Prof. Dr. Ernst Binz (auth.)

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**Example text**

I Hom Note that any homomorphism in see [ Ri ], meaning that Hom A A has norm less than or equal to i, is in the polar of the unit ball of A. This means: CoroIlar~ 30 If A is a normed HOmcA A - a l g e b r a then is a compact topological space, which carries the topology of pointwise convergence. To collect the results on compact c-embedded spaces we state [ Bi 4 ]: Theorem ~I For any c-embedded convergence space If Cc(X) X the following are equivalent: (i) X is compact. (ii) X is compact and topological.

T be the coarsest We will first ~ 9 f ~ C(X) The final topology on e-admissible show that the members of the zero function On of the C(X) in CT(X) topo- of the neighbor- is absorbant. we choose the natural To this vector induced by the inclusion space map is 28 m-admissible. tinuous, Hence the inclusion map of implying that neighborhoods m-admissible for some we find to each U s the interior in C(X) ~(o). Wp of extend to f E C(X) into 9 f T a neighborhood Vp in such that in [--3 Wp ps Thus . is con- Since Hence all functions Since CT(X) f.

If for a c-embedded space topological, then X X the convergence algebra Cc(X) is is by corollary 29 a locally compact space. Thus locally compact c-embedded spaces can be described by their associated convergence function algebra as expressed by M. 1 ]: 52 Theorem 32 For any c-embedded If Cc(X) space X the following (i) X is locally (ii) Cc(X) are equivalent: compact. is topological. carries a topology, then it is the topology of compact convergence. Let us now investigate locally compact of view.