Dual A * -Algebras of the First Kind

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 74, Number 2, May 1979 DUALA»ALGEBRASOF THE FIRST KIND DAVID L. JOHNSON AND CHARLES D. LAHR Abstract. Let A be an A '-algebra of the first kind. It is proved that A has property P2 of Maté if and only if A2 is dense in A if and only if A possesses an (operator-bounded) approximate identity. Further, it is shown that an A '-algebra of the first kind having property P2 is a dual algebra if and only if it is a modular annihilator algebra. As applications, these results are used to strengthen certain theorems about Hilbert algebras. 1. Let A =(A,\\- ||) be an ^»-algebra, and let 21= (91,| • |) be its C*- algebra completion with respect to the auxiliary norm | • | [11, p. 181]. If A is a (dense two-sided) ideal of 9Í, then A is called an A *-algebra of the first kind. In this case, there is a constant K > 0 such that ||wa||< AT|w|||a||, ||úrw||< K\w\ \\a\\, for all a in A, w in 91 [1, Proposition 2.2, Theorem 2.3]; thus, A is a Banach 2i-bimodule. In this paper, several characterizations of Mate's property P2 [9] are given for A *-algebras of the first kind, and it is shown that such an algebra is dual if and only if it is modular annihilator. These results are used to strengthen some theorems in [16] concerning Hilbert algebras. 2. Let A be a Banach algebra with Banach space dual A *, and let A 0 A * denote the projective tensor product of A and A* [12, p. 93]. For a in A and/ in A*, define / * a in A* by </ * a, b} = </, ab}, all b in A. It is immediate that A* is a right Banach A -module with respect to this product. Moreover, by the universal property of the projective tensor product, there is a unique continuous linear map B: A 0 A* -> A* such that B(a 0 f) = f * a. For a general tensor ? = 2 a*®/* *=i in A 0 A*, where ak G A, fk G A*, and 2"_,||afc|| ||/*|| < +oo, it follows that B(Ç) = 2^=1/t * ak. It also follows from the universal property of projective tensor product that there is a unique continuous linear map E: A 0 A* -» C such that E(a 0 f) = </, a}; more generally, for a tensor f in A 0 A* as above E(Ç) = 2£Li<fk, ak). The Banach algebrad is said to have Received by the editors April 25, 1978. AMS (MOS) subjectclassifications(1970).Primary 46H20,46L05,46L20;Secondary46H25, 46K15. Key words and phrases. Approximate annihilator algebra, Hilbert algebra. identity, multiplier, A '-algebra, 311 dual algebra, modular © 1979 American Mathematical 0002-9939/79/0000-0221/$02.00 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use Society 312 D. L. JOHNSON AND C. D. LAHR property P2 if ker(2?) C ker(F); in other words, if 00 00 k=\ k=\ S Nlll/*ll< +«> and 2 fk*ak= 0, then 2?_ ,</*, ak) = 0 [9], [14, Theorem 2.3]. For a Banach algebra A to possess property P2, A must have a right approximate identity [7]. In this paper, every approximate identity {ux) will be two-sided (i.e., ||«Aa - a\\ ->0, ||a«A - a\\ ->0, for all a in A). This is no restriction since the algebras under consideration all have continuous involution. An approximate identity {ux) for A is said to be operator-bounded if sup{||HAa||, ||atix||: a E A, \\a\\< 1} < +oo. Finally, A2 will denote the linear span of products ab, where a, b E A. Theorem I. If A is an A*-algebra of the first kind, then the following statements are equivalent: (a) A 2 is dense in A. (b) A has property P2. (c) A has an approximate identity. (d) A has an operator-bounded approximate identity. Proof. The implications (d) => (c) => (a) are immediate; therefore, it suffices to show that (b) =>(a) =>(d) =>(b). ((b) =>(a)). Suppose that A has property P2. Then, by [14, Theorem 2.3], there exists a net {ux} in A such that </, aux} -* </, a}, for all a in A, fin A*. It follows that no nonzero element/of A* can annihilate A2; thus, (a) holds. ((a)=>(d)). Let 31= (91,| • |) be the C*-algebra completion of A - (A, || • ||) in the auxiliary norm | • |. Because A is a dense ideal of 91,it contains a (two-sided) approximate identity {wA}for 91 such that u* = ux and ||i7A|| < 1, for all X [3, Proposition 1.7.2]. Now, since A is a Banach 9I-bimodule, by the Hewitt-Cohen factorization theorem [4, Theorem 32.22], 91-^4 = {wa: w E 91, a E A] (resp., A ■91) is a closed linear subspace of A. However, A2 C 91• A n A ■91,and A2 is dense in A by hypothesis. Therefore, A = 91• A = A • 91, and it follows easily that {ux} is an operator-bounded approximate identity for A. ((d) => (b)). Suppose that A has an operator-bounded approximate identity {ux}. Then standard estimates show that, if 2~_i||afc|| ||/J| < + oo, then 00 00 k=l k=l in A ® A *. Consequently, if 2"_ xfk * ak = 0, then OO 00 / 00 \ 2 </*.«*>= lip 2 </*»4tMA> = »m ( S /* * <**> ux) = 0; *=i A *=i A \*=i hence, A has property P2. □ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use / DUAL A ""-ALGEBRASOF THE FIRST KIND 313 3. A Banach algebra A is said to be modular annihilator if every modular maximal left (right) ideal of A has a nonzero right (left) annihilator (see [2]), and is said to be dual if every closed left (right) ideal in A is an annihilator ideal [11, Definition 2.8.1]. The former notion is purely algebraic, while the latter is topological; yet, a C*-algebra is modular annihilator if and only if it is dual [15, Theorem 4.1]; [2, Example 4.1]. The next theorem shows that the same is true of A *-algebras of the first kind possessing property P2. If A is a semisimple Banach algebra, then ML(A) will denote the Banach algebra of (automatically) continuous left multipliers of A. Finally, the bidual 91** of a C*-algebra 91 is equipped with a product, extending that of 91,with respect to which it is a von Neumann algebra [3, §12.1]. Theorem 2. Let A be an A*-algebra of the first kind with A2 dense in A, and let 91 be the C*'-algebra completion of A. Then the following statements are equivalent: (a) A is a modular annihilator algebra. (b) 91is a modular annihilator algebra. (c) A is a dual algebra. (d) 91is a dual algebra. (e) ML(A) is algebra isomorphic to 91**. (f) ML(%) is algebra isomorphic to 91**. Proof. The implications (c) =*> (a), (d) =>(b), (b) =>(a), and (a) =>(d) are either immediate or well known (see [2], [15]). The equivalence (d) <=>(f) may be found in [10], and the implication (c) => (e) is the content of [14, Theorem 4.2, p. 264]. To complete the proof, the implications (a) => (c), (e) =* (f) will be established. ((a) =>(c)). This follows from [13, Corollary 4.4, p. 426] in view of [1, Theorem 4.2, p. 6] (a slip is made in [1, Proposition 3.3] in not assuming that A =%-A). ((e) => (f)). First, since 91 has a bounded approximate identity of norm one, Af¿(9í) is isometric and algebra isomorphic to a closed subalgebra of 9Í**. But the hypothesis (e) implies, since 91 is the closure of A in 91**, that every element of 91** determines an element of ML(W). □ 4. If A is a replete Hilbert algebra (for definitions, see [16]), then in the so-called Rieffel norm || • \\r,A is an ^l*-algebra of the first kind such that A2 is dense in A [5, Theorem 4.1]. As a result, Theorem 2 applies, and yields the following stronger versions of Theorems 3.7 and 4.3 of [16]. Proposition 3. Let A be a replete Hilbert algebra with C*-algebra completion 91. If 91 is dual, then A (as a Hilbert algebra) is dual and has dense socle. Proof. By Theorem 2, the Banach algebra Ar = (A, || • ||f) is a dual algebra and, as a result, Ar has dense socle. Since the map A -» A is a norm-decreas- License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 314 D. L. JOHNSONAND C. D. LAHR ing bijection, the socle of A is also dense in the Hilbert algebra A. Finally, [16, Theorem 3.7] shows that A is a dual Hilbert algebra. □ Proposition 4. Let A be a replete Hilbert algebra with C*-algebra completion 91. Then 91 is dual if and only if A is modular annihilator. Proof. Immediate from Theorem 2(a), (d). □ Finally, since every full (i.e., maximal) Hilbert algebra is a replete Hilbert algebra, Proposition 4 can be related to an example in [8], where it is shown that the trace class t(A) of a full Hilbert algebra A, regarded as a subspace of 9Í*, need not be dense in 91*. In fact, the following is true. Corollary 5. Let A be a full Hilbert algebra with trace class r(A) and C*-completion 91. Then r(A) is dense in 91* // and only if A is a modular annihilator algebra. Proof. In [6], it is shown that t(A) is dense in 91*if and only if 91is dual; hence, the result follows from Proposition 4. □ References 1. B. A. Barnes, Banach algebras which are ideals in a Banach algebra, Pacific J. Math. 38 (1971),1-7. 2. _, 657-665. Examples of modular annihilator algebras, Rocky Mountain J. Math. 1 (1971), 3. J. Dixmier, Les C*-algèbres et leurs représentations, Cahiers Scientifiques, fase. 29, Gauthier-VL: js, Paris, 1969. 4. E. Hewitt and K. A. Ross, Abstract harmonie analysis. II, Springer-Verlag, New York, Heidelberg, Berlin, 1970. 5. D. L. Johnson and C. D. Lahr, Multipliers and derivations of Hilbert algebras (to appear). 6._, The trace class of an arbitrary Hilbert algebra (to appear). 7. C. A. Jones, Approximate identities and multipliers, Ph. D. Dissertation, Dartmouth College, 1978. 8. M. R. W. Kervin, The trace-class of a full Hilbert algebra, Trans. Amer. Math. Soc. 178 (1973),259-270. 9. L. Maté, The Arens product and multiplier operators, Studia Math. 28 (1967), 227-234. 10. E. A. McCharen, A characterization of dual B*-algebras, Proc. Amer. Math. Soc. 37 (1973), 84. 11. C. E. Rickart, General theory of Banach algebras, 1974 reprint, R. E. Krieger, Huntington, New York, 1960. 12. H. H. Schaefer, Topological vector spaces, Springer-Verlag, New York, Heidelberg, Berlin, 1971. 13.B. J. Tomiuk,ModularannihilatorA*-algebras,Canad. Math. Bull.15 (1972),421-426. 14._, Multiplierson dual A*-algebras,Proc. Amer. Math. Soc. 62 (1977),259-265. 15.B. Yood, Ideals in topologicalrings,Canad. J. Math. 16 (1964),28-45. 16. _, Hilbert algebras as topologicalalgebras, Ark. Mat. 12 (1974), 131-151. Department of Mathematics, (Current address of C. D. Lahr) Dartmouth College, Hanover, New Hampshire 03755 Current address (D. L. Johnson): Department of Mathematics, University of Southern California, Los Angeles, California 90007 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use