By de la Harpe P., Jones V.

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A character on A is a linear map : A ! distinct from zero such that (ab) = (a) (b) for all a b 2 A: The set of all characters on A is denoted by X (A): If A has a unit, observe that (1) = 1 for any 2 X (A): If A has no unit, any character on A extends uniquely ~ Moreover X (A~) is to a character ~ on A~ de ned by ~( a) = + (a) for all ( a) 2 A: ~ naturally identi ed to the union of X (A) and of the character ( a) 7! of A: If A is a commutative Banach algebra with unit, any character on A satis es (*) sup j (a)j 1 a2A kak 1 for all a 2 A and in particular any character on A is continuous.

The subspace F (H) is obviously a two-sided ideal. It is self-adjoint by the previous lemma. 4. Recall of vocabulary. Let X be a topological space. A subset Y of X is relatively compact if its closure Y is compact. Assume moreover that X is a metric space with distance d: A subset Y of X is precompact if, for every real number > 0 there exists a nite subset S of Y such that Y x S B (x ) where B (x ) denotes the open ball of center x and of radius : Inside a complete metric space, a subset is relatively compact if and only if it is precompact.

19. Proposition. (i) Let a 2 B(H H ) let ( j )j J be an orthonormal basis of H and 0 let ( k )k K be an orthonormal basis of H : Let (ak j )k K j J be the resulting J -times-K matrix, where ak j = h k j a j i: The three following conditions are equivalent 2 0 2 2 2 a is a Hilbert-Schmidt operator, X 2 ka j k < 1 Xj J 2 k2K j 2J If they hold, then ja j < 1: kj 2 kak = X ka k 2 j 2 = j 2J X k2K j 2J ja j kj 2 and the adjoint a of a is also a Hilbert-Schmidt operator such that ka k = kak kak : 2 2 Proof.