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NOTES and REMARKS Chapter I.I. the Concept of a Bounded NOTES AND REMARKS Chapter I.i. The concept of a bounded approximate identity goes back to the earliest studies of the group algebra LI(G) [282]. Approximate identities in LI(G) are discussed in a systematic way in A. Weil's book [281], especially pp. 52, 79-80, and 85-86, and they have since become a standard tool in harmonic analysis [131], §28 (called bounded approximate unit); [80], Chapter III (called approximate identity); [147], Chapter I (called summability kernel); [220], §6 (called bounded multiple units). For applications to group representations it soon became necessary to consider C*-algebras without identity element. In order to discuss such algebras I. E. Segal constructed for every C*-algebra an approximate identity bounded by one [243], Lemma i.i. Many results about a C*-algebra without an identity element can be obtained by embedding such an algebra in a C*-algebra with identity element, called adJunction of an identity (see [222], Lemma (4.1.13) or [70], 1.3.8). But some problems, especially those which involve approximate identities, are not susceptible to this approach. Therefore, the approximate identity is the main tool in Dixmier's book [70] (called unite approchee in the earlier French editions) to carry through all the basic theory of C*-algebras. In recent years many results known for C*-algebras or for Banach algebras with identity have been extended to algebras with an approximate identity. The mainspring for much of this work was the Cohen-Hewitt 238 NOTES AND REMARKS factorization theorem for Banach modules [131], §32. The elementary results in I.l are very familiar. For Proposition (1.2) see H. Reiter [220], §6; for Lemma (1.4) see P. G. Dixon [71], Lemma (2.1); for Proposition (1.5) see J. Dixmier [70], 1.7.2. 2. The results relating left, right, and two-sided approximate identities are due to P. G. Dixon [71]. Our Lemma (2.1) is a reformu- lation of Dixon's Lemma (3.1) which contained an ambiguity in its statement. The authors thank Professor C. R. Combrink for his insights on this lemma. For Proposition (2.5) see [71], Proposition (4.4) and also C. R. Warner and R. Whitley [280], p. 279. 3. It is well known that if a normed algebra A has an identity then there is an equivalent algebra norm on A such that the identity has norm one [30], 1.2.1, pp. 13-14; [152], p. 52. The analogue for approximate identities has been studied by P. G. Dixon [71], [72]. For Lemma (3.2) see F. F. Bonsall and J. Duncan [27], p. 21; for Theorem (3.3) see also W. B. Johnson [144], p. 309. Theorem (3.4) can be found in Dixon [72] as Theorem (2.3), and Theorem (3.5) is due to A. M. Sinclair [250], Theorem 8 (the version given here is a nonseparable generalization of Sinclair's theorem to normed algebras, see [72], Theorem (2.4)). 4, 5, 6. The results of these three sections have been taken from Dixon [72]. 7. The result about the existence of approximate identities in quotient algebras can be found in H. Reiter [220], §7, Lemma 2 and Lemma 3. 239 NOTES AND REMARKS Problem. Let A be a normed algebra and let I be a closed two- sided ideal in A. If I and A/I have left approximate identities, does A have a left approximate identity? Example (7.2) is due to P. G. Dixon (private communication). 8. For the general theory of tensor products of normed linear spaces see R. Schatten's book [241] or A. Grothendieck's monograph [119]; for the tensor product of Banach algebras and Banach modules see [29], [102], [103], [104], [172], [120]. The first result concerning identities in A ® B was that of B. R. Gelbaum [i00], Theorem 4: Let A and B be commutative semisimple Banach algebras. Then A ® B has an identity if and only if A and B have Y identities. L. J. Lardy and J. A. Lindberg proved this result for any "spectral tensor norm" [163]; for an elementary proof see R. J. Loy [185]. The fact that A ® B for any admissible norm I I.II~ has a bounded approximate identity if A and B have bounded approximate identities seems to be well known; it is used implicitly by K. B. Laursen in [173], Theorem 2.2. The first systematic study of approximate identities in tensor products of Banaeh algebras was made by R. J. Loy [185]. D.A. Robbins [230] and J. R. Holub [133] improved his results. The short proof of Theorem (8.2) in the text was communicated to us privately by B. E. Johnson. 9. The need for the more general concept of approximate units in a normed algebra first arose in connection with the study of Wiener-Ditkin sets [218], Chapter 7, §4, especially §4.10, and became more apparent in 240 NOTES AND REMARKS H. Reiter's lecture notes [220]. Proposition (9.2) was motivated by M. Altman's study of contractors [8], [i0], [ii]. For a systematic study of contrac6ors see Altman's recent book [13] on the subject. The most important and very useful result is the equivalence of the existence of bounded approximate units and the existence of a bounded approximate identity in any normed algebra (Theorem (9.3) and Theorem (9.4)). The present sharp result was obtained successively by H. Reiter [219], [220], §7, Lemma i; M. Altman [I0], Lemma i; and J. Wichmann [284]. The study of pointwise-bounded approximate units was taken up by Wichmann in [147]; for Theorem (9.7), see also T. S. Liu, A. van Rooij, and J. K. Wang [182], Lemma 12. Problem. Let A be a Banach algebra with pointwise-bounded left approximate units. Does A have a bounded left approximate identity? Problem. Let A be a Banach algebra with left approximate units. Does A have a left approximate identity? Problem. Does Theorem (9.3) generalize to Banach A-modules? (Example: Let X be a Banach space and let B'(X) be the Banach algebra of compact operators on X. Then for every x e X and every e > 0 there exists a compact operator u e B'(X), depending on x and e, with llull i such that llx - ux]l < e. Does there exist a bounded net {el}le A of compact operators on X such that lim elx = x for all x e X?). lea The appropriate reformulation of statements and proofs concerning 241 NOTES AND REMARKS approximate identities for approximate units is left to the reader. For approximate units in Segal algebras and the method of linear functionals see Chapter III, §32. i0. First results about approximate identities in normed algebras which do not consist entirely of topological zero divisors appeared in R. J. Loy [185], Proposition 1 and Proposition 3. These results have been extended in this section by the second author, with the aim to solve for such algebras the problems mentioned above about approximate units. Some illustrative examples and counterexamples would be desirable. ii. This section was motivated by the interesting paper [199] of J. K. Miziolek, T. M~idner, and A. Rek on topologically nilpotent algebras. Proposition (11.6) extends [199], Proposition 2.4. 12. The study of C*-algebras without an identity element is more than just a mildly interesting extension of the case of a C*-algebra with iden- tity. The main tool is the approximate identity which such algebras have. In fact, as the proof of Theorem (12.4) shows, in any left ideal I of a C*-algebra there is an increasing net {e%}%c A of positive operators bounded by one such that {e%}%e A is a right approximate identity in the norm closure T of I. The construction of such an approximate identity is due to I. E. Segal [243], Lemma i.i. It was proved by J. Dixmier, Traces sur les C*-algebres, Ann. Inst. Fourier 13 (1963), 219-262, that this approximate identity is increasing; see also [70], 1.7. To see the importance of approximate identities in the extension of the Gelfand-Naimark theorem for noncommutative C*-algebras (with weak 242 NOTES AND REMARKS norm condition llx*x]] = IIx*ll.llxlI) with identiny to the case of C*-algebras without identity see R. S. Doran and J. Wichmann [73]. Also, see the original paper by B. J. Vowden, On the Gelfand-Naimark theorem, J. London Math. Soc. 42 (1967), 725-731. In the case of a commutative C*-algebra C (S) the following question o arises: what do restrictions on the approximate identity imply about the spectrum S of C (S) and vice versa? Along this line, Proposition (12.1) o characterizes compactness of the spectrum; the characterization of o-compact- ness of S in Proposition (12.2) is due to H. S. Collins and J. R. Dorroh [52], Theorem 4.1. We will discuss later (~28) the characterization of paracompactness (every open cover has a locally finite open refinement) of S in terms of approximate identities. It is interesting to ask similar questions for an arbitrary C*-algebra A and the space S = Prim(A) of primitive ideals furnished with the Jacobson topology (or the space A of classes of nonzero irreducible representations of A) [70], Chapter 3. Recall that a *-algebra is symmetric if for every x, the spectrum o(x*x) is a subset of the nonnegative reals. Problem. Does there exist a commutative, semisimple, symmetric Banach *-algebra with o-compact carrier space but no bounded sequential approximate identity? The construction of a sequential increasing abelian approximate identity for C*-algebras with a strictly positive element (see (12.9)) is due to J.
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