TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 180, June 1973 P COMMUTATIVEBANACH^ALGEBRASt1), (2) BY WAYNE TILLER ABSTRACT. Let A be a complex *-algebra. If / is a positive A, let /, = ¡x £ A: f{x * x) = 0| be the corresponding where the intersection P-commutative and examples commutative. the extend results 2 if and only 1. Introduction. speaking, of this called erties Banach The structure which bras, some prove discuss several -algebra -algebras tative we shall theorems A Banach functional -algebra Presented and, in revised -algebras are on If A is a pure the spectral radius positive I. Murphy on A implies a nonzero January properly for P-commutative -al- the class the notation P-commutative ideals, and then Banach on A is automatically for commutative [5] gave show identity, a different that Banach proof of a multiplicative linear 17, 1972; received which if A is a P-commu- then the existence the existence and alge- and quotients, functional result prop- algebras. if A is a P-commutative We also positive theory. Banach contains In §2 we introduce the same with an apptoximate noncommutative In §3 we define that is, generally many of the well-known true subalgebras, involution. to the Society, them * -algebras which see, remain is as follows. [9] proved on A; and in fact, Among of noncommutative -algebras, as we shall of the involution. functional Banach a new class = A, then every continuity P- A is continuous. then than the corresponding In §4 we show with continuous zero positive of commutative concerning examples. are *-algebras functional If A is symmetric, use throughout. N. Varopoulos did not require positive Banach Banach satisfying continuous. a nonzero is P- which Banach on A is called *-algebra seminorm. of the paper terminology then -algebras; of commutative Then *-algebras. functional is to study P-commutative of commutative Banach positive understood paper A. *-algebras every The theory much better The purpose on commutative for P-commutative if it is multiplicative. algebra Every of noncommutative for commutative then identity, in A is a continuous gebras, -A, on left ideal of A. Set P = I I /,, functionals x, y £ A. are obtained known If A an approximate state all positive are given Many results following: A has over if xy —yx £ P for all commutative which is functional functional of a non- linear is a pure by the editors February state 22, if 1972 form, July 24, 1972. AMS(MOS)subject classifications (1970). Primary 46H05, 46J99; Secondary 46K99. Key words and phrases. Banach *-algebra, positive multiplicative linear functional, symmetric *-algebra. (1) Texas (2) functional, *-representation, The contents of this paper form a portion of the author's doctoral dissertation Christian University under the direction of Professor Robert S. Doran. This research was partially supported by the National Science at Foundation. Copyright © 1973, TVmericanMathematical Society 327 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 328 WAYNETILLER and only which if it is multiplicative. are symmetric. algebra is a continuous tion of symmetry -algebra vix -algebra to have by one. A by adjoining and norm one, in [7]). ideal R ). iated "MLF". If / is a positive of A; that The words g (x v(A / denote on A, we set "pure" and obtained from radical of A, of the kernels the the radical functional x) < f(x will (R is called linear be called a con- be two-sided is, the intersection we let is, the Jacobson spaces algebras. identities the algebra "multiplicative / on A will g on A satisfying radius x—> x ; that will by A (or / .) denote convenience we write functional Banach linear algebras identities of A on Hilbert larly, A positive / complex In normed we denote We let -representations For typographical in such an the assump- if its spectral an involution two. and approximate -algebra, R (or R .) the reducing that an arbitrary associative with of period an identity. all irreducible only an algebra If A is a show that is P-commutative consider anti-automorphism be assumed bounded mean will we show -algebras radius x. We shall we will Banach that the spectral An example Finally identity for all 2. Preliminaries. jugate-linear seminorm. be dropped. x) < (x) P-commutative among other things, algebra cannot with an approximate satisfies By a In §5 we study We show, functional" of -radical of A (simi- will be abbrev- I, = \x e A: f (x x) = 0\. if for every x) for all x £ A, it follows -algebra. A continuous positive that functional g = Kf for some 0< kK 1. Now let A will A denote be called trum of an element remark that tor exists r) £ II such continuous, that show / that (x) = f (y then functional book and let [7] will not require of J. Ford's immediate P = I I /,, o . (x)) the spectral radius every x e A. root lemma We / on H and a cyclic that vec- / is Hermitian, any results of local we use continuity of [3]. and some the intersection of x. functional In particular, the assumption square / on the spec- to be representable space we remark consequences, where positive / on A is said Finally, functional (or 77 of A on a Hilbert f (R) = 0. positive by ct(x) (or v . (x)) f (x) = (?t(x) r¡ \ rf) for every by virtue examples. is taken over Let A be all positive on A. Proposition Prool. We denote identity, -representation 3. The definition, -algebra, = 1. [8]; a positive a and satisfies the involution functionals ||/|| x in A, and by vix) from C. Rickart's a if if A has an approximate A is representable if there a Banach a "state" Since 3.1. P is a two-sided P is the intersection P is a right xy) ideal for each ideal. of a family If / is a positive x £ A. Then in A. / of left functional is a positive License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ideals, it suffices on A and functional to y 6 A, set on A, and if P-COMMUTATIVE BANACH *-ALGEBRAS x £ P, we have right ideal. The /[(xy) x xy) = / (x x) = 0. Hence xy £ P and P is a Q.E.D. following Definition every (xy)] = fiy 329 definition 3.2. was A -algebra suggested to the author A is said by Professor to be P-commutative R. S. Doran. if xy-yx £ P for x, y £ A. Proposition 3.3. Lez" A be a P-commutative -algebra. Then A is P-commu- tative. Proof. -(y, It is clear that a) (x, A) = ixy-yx, P-commutative. 3.4. -homomorphism Proof. Let <7j(x)e -cb(y) Let since A is P-commutative. A be a P-commutative of A onto B. Then / be an arbitrary tive functional thus, 0) £ P . Then (x, A)(y, a) Therefore, A is Q.E.D. Proposition a PA C P . Now let (x, A), (y, a) £ A£. P_. Hence, It follows positive functional if 0 (x), cbiy) on B. 0 = if ° (b) ix are arbitrary e <tj(P .) C P„. from Proposition commutative B a -algebra, and d) B is P-commutative. on A ; so if x e P A, then cbix) = 0(xy-yx) -algebra, Therefore, 3.4 that Then f ° cb is a posi- x) = f[cb(x) elements cb (x)], and of B, then q>ix)cb(y) B is P-commutative. if / is a -ideal in A, then Q.E.D. A/1 is a P- -algebra. For the remainder of this paper, unless stated otherwise, A will denote a a. Banach -algebra. Since for every Banach -algebra A, it is true (see [7, 4.6.8, p. 226]), it is true that P-commutative, then and reduced identity, and (i.e., then A/R is commutative. R = iOi), then every positive A is P-commutative need an approximate then that a closed not be P-commutative. tion to illustrate Proposition functional this fact. 3.5. However, }e j. / representable that if A is if A is P-commutative Now if A has on A is representable, if A/R an approximate and hence, P = R is commutative. -subalgebra of a P-commutative We will give we have Lez: A be a P-commutative identity R = I 1/,, In particular, A is commutative. if and only It is not surprising -algebra that P C R. So we see immediately // B is a closed an example the following symmetric later Banach in this sec- result. Banach -subalgebra -algebra of A containing with ¡e B is P-commutative. Proof. notation ducing o We adjoin ßj ideals in this identities proposition R e of A e and to A and B to obtain only so that R, \ of B, \J). trie [7, 4.7.9, p. 233], and ß j is a closed A and we may distinguish Now B, between A e is P-commutative -subalgebra License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use (we use the the re- and symme' of A . Since both A and ¡, 330 WAYNE TILLER B possess approximate identities, all positive functionals are representable, and it follows that PA = RA = Re and Pß = Rß = R v So by [7, 4.7.19, p. 237], we have that Pß = R{ = By O Rg = ßJ n P^ = ß n P^; it follows that mutative. ß is P-com- Q.E.D. A related result Proposition is given 3.6. 1 is a closed Let -ideal next. A be a P-commutative of A having symmetric an approximate Banach identity, then -algebra. If I is P-commuta- tive. Proof. it follows We have that P. C R . = / . and P. = P. = /,. that /, = / O JA. / H P . is obvious; We now give mutative, hence some 3.7. by giving We next Example norm ||-|| ments 3.8. Let norm Let / be a positive set a = (x, 0). arbitrary, a £ P. functional -algebra having use algebra need -subalgebra tation of A has no nonzero We next it is reduced. which implies noncommutative Furnish Banach We now show A is that element a) = 0; hence (x, y)eP an identity e, then with ele- algebraic = (y , x ). Then A is P-commutative. element in A , and a £ I, and since of A of the form that algebra We denote (x, y) x be an arbitrary /(zz |. A with pointwise , and involution on A, let every if A on A, choose Example 0) £ A = P A = ¡(x, y): x, y e A + |y|| in A, it follows 3.8, of A. Then |/(«)|2 tive Let a = 0 implies Similarly, (i.e., (0, y) is in for every / was P. x, y £ A ; i.e., But, A = A is P-commutative. In Example functional a -algebra is semisimple. be a semisimple, x—> x . and semisimple. P is an ideal Therefore, which (x, y) or a, b, etc. Then Banach is noncommutative. ||(x, y)|| = ||x|| A is noncommutative P. A of A by either operations, since A an example and involution of noncomr P e = P e = A. Now if (x, A), (y, a.) £ J a) (x, A) = ixy-yx, Obviously, construct a few examples radical /J A. = P A. = A).' Then, ' for A e , we have that P-commutative. P, C -algebras. A be a noncommutative A , then (x, A) (y, a)--(y, / is an ideal, The inclusion Q.E.D. We begin Banach Let Since P, 3 / C\ PA. / is P-commutative. examples. Prcommutative Example Thus, we obtain Indeed, a £ A, and let 3.8 to show that let of a Banach / be a positive 1 = (e, e) denote the identity = 0 and so f = 0 on A. a closed -subalgebra of a P-commuta- not be P-commutative. Let Now let of A. functionals. = |/(1 -a)\2 < f(l)f(a*a) Example 3.,9. positive an arbitrary A is an example A be as in 3.8 above ß = í(x, x): x e A j. We show on some Hilbert that space but with the added Then ß is a closed B is not P-commutative. H. Let assumption noncommutative n be a -represen- Then the map 77R: B —• B(H) License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use that defined by P-COMMUTATIVE BANACH *-ALGEBRAS 77g [(x, x)) = 77(x) is a (x, x)eker(/7R). -representation Since commutative, A' it follows Example 3.10. with continuous Let A is noncommutative B is symmetric ments of the form x, y £ A, i.e. identity e. We show P, z + r, where every z £ Z, r£RR, P„ is non- by Z and P„. A is P-commutative. = RA = A C\ PR = PR. (z x + r x) (z, + r2)-U2 in A and since that -algebra that Now + r 21 ^z l + rl^ = rlr2~r2rl element of A is the limit it follows that xy-yx e of ele- £ R . for every A is P-commutative. We next shows Then P . is closed B is non- Banach of B generated and e £ A, we have that z2 + r 2£ A. Since since Z, and assume -subalgebra and has if and only if Therefore, symmetric e, and center A be the closed Since + t., identity x £ ker(7r) be P-commutative. B be any noncommutative, Then P .. ß is reduced. B cannot involution, commutative. let z is reduced, that Let of B on H and 331 give that, an example in general, showing nothing that P is not always can be said about equal to P. the relationship It also between / and P. 2\a Example 3.11. \/n\ (it will < » the usual tion 2a Let algebraic A be the set be understood operations, zn—>2â~ z". 7Z that the power sum runs norm defined Then 72 of all fotmal by from \\2a A is a commutative, series, 2 a z", n = 1 to oo). zn\\ = 2 \a radical \/n\, Banach Then / is clearly there implies linear, exist / is positive series since in A for which a f[2ä~ / -algebra, & zn) i2a ' so zn) = «y zn)] -- \a A 0, it follows A that I,/ > 0. A which P / A. A. Multiplicative A is a Banach of pairs and power Give and involu- / = P = A. We show that P / A. Consider f: A — C defined by f(la Since satisfying linear -algebra, of elements Theorem. dominate from Assume A. A a continuous functionals we denote and continuity by A I. Murphy the set [5] proved = A and let every nonzero positive of positive of all finite the following nonzero functional. If of products result: positive Then functionals. sums every functional on A positive functional on A is continuous. then As a corollary, Murphy every functional positive commutative being algebras. in showing Theorem then every Prool. since every (x if A is commutative x) = / follows iu functional / be a nonzero Murphy's and satisfies We extend this essentially, A2 = A, result the main to Pdeviation u). // A is a P-commutative positive Let / that on A is continuous. The proof that 4.1. proved Banach -algebra satisfying A = A, on A is continuous. positive x in A can be written functional on A. in the form x = 2"_ License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use Then , X.x .x. / is Hermitian (see [5]). For WAYNE TILLER 332 every u £ A, let / 3.1. Then / every u £ A, then /(A3) = 0 which 0, and since lemma that be the positive is continuous |/(z/*x*y)|2 is false / = |a| then provides (x x) = / iu / is well known defined by functional < fiu *x *xu)fiy since A^ = A. in the proof of Proposition v £ A u) for every that the linear x £ A. space (x + I, \y + IA = fiy that such together /, that that root in A. product It (• | •) Furthermore, zzx + /, = xzz + /, for every 4 Now we show is a left ideal space. / square u. with the inner x), is a pre-Hilbert it follows Ford's v v = e-u that zz, x, y £ A; hence, u £ A such \\u u\\ < 1. that Recall A/1 Now if fu= 0 fot *y) = 0 for every So we may choose / , we may assume an element A is P-commutative, defined on A (see [7, 4.5.4, (ii), p. 215]). since x e A. Therefore, we have / (x x) = fiu x xu) = ixu + I Axu + I A = (zzx + IftAux + IA = fix Now for every x £ A, if-f) (x*x) = f(x*x)-fu(x*x) = fix*x)-fxiu*u) = f[x*íe-u*u)x] since / is positive an application on A and of Murphy's continuous. The following lemma Lemma 4.2. Let the identity = f[ívx)*ívx)] > 0 in A . So / dominates we see that It is well every positive / , and by functional on A is on H. Let Then p. 28] for the case of the proof involution. -algebra We therefore with approximate of A on a Hilbert Then A that it omit the proof. identity space lim 77(ea) = 1, where when in [2] shows je H, and let the limit ¡. Let 1 de- is in the on B (H). known that 4.3. \ea\. with arbitrary -representation topology [2, 2.2.10, modification A be a Banach is a pure state Theorem known A slight operator operator identity is well algebras v be a nondegenerate functional A is an ideal theorem, involution. is true for Banach note = fíx*v*vx) Q.E.D. has isometric strong u ux) = f x iu u). on a commutative Banach -algebra a nonzero positive if and only if it is multiplicative. A be a P-commutative a nonzero positive functional Banach -algebra with approximate f on A is a pure state if and only if it is multiplicative. Proof. f(R)= Clearly that Let / be a pure state 0 and so we may define / is linear 11/11< ||/ ||. and positive, Now positive on A. a function Since and it is easy scalar A has an approximate identity, / ': A/R —> C by / ' (x + P) = fix). multiples to check that of pure positive License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use it is also pure and functionals are 333 P-COMMUTATIVE BANACH "-ALGEBRAS pure positive A/'is functionals; a pure state multiplicative able, on A/R. which and space f(x)/, ||/|| n £ H. since the first Now assume define / that state. 77 is a a. that A< / limf(ea) positive [7, 4.4.12, relative MLF on A. If we choose which implies MLF Again on A. then, f(R) = 0, so we on A/R x £ A such limf(ej and, that irreducible ||/|| then of A must the set of scalar If we furnish case, M with the M is a locally M is compact. functions statej the set of irreducible (f £ M). compact If we define on M which x—> x is a norm decreasing But we set is just M is precisely as in the commutative > 1. Q.E.D. -representation image hence, f (x) /= 0, then = 1; hence identity, of each R = flker(/) Cn(zVI) (the continuous then suchthat \\r]\\2 < ||/|| = on A\ = ¡/: / is a pure of every the commutant If A has an identity, x—> x of A into = fix), the image Therefore, -topology, space. positive p. 211], we see that of A. weak Hausdorff by x(f) Since and since -representations xeA (ea) which implies and multiplicative and has an approximate M = \f : j is a nonzero operators = ||771| ; A= 1 and so / is multiplicative is positive / is pure. If A is P-commutative be commutative, is on the 1. But lim/(ea)= 11/11 S 11/'II= 1 implies u/11= 1, and therefore / is a state. (M may be empty). A/ lima/(ea) Now if we choose 1 since that Now / is represent- -representation 4.2, we see = (A/) (x) lim(A/) 1/A> = 1; thus Then = fix) it follows on A. that of the theorem. Therefore, l'im/(xea) A, 0 < A < 1, such is commutative, rj), where / is a nonzero as before. is a pure f(x)= half a scalar by Lemma = lim(A/)(xea) 1/A = lim/(ea) proves A/R < 1 for every 1 or lim/(ea)= 1, and hence This exists A/ is multiplicative Then ||ej| 0, then (A/)(x) lim(A/)(ea)= that f(x) = ivix)r\\ H and ||z7||2< there Since implies so we may write Hilbert hence, vanish -homomorphism. the map at infinity) It is injective if and only if A is reduced. 5. Symmetric, commutative bras have already -algebras 5.1. Let in A is primitive Proof. in A.. Then Banach which been obtained Proposition ideal P-commutative Banach The "if" since (see A be symmetric implies modular in A. l/R is maximal section we study for such and P-commutative. P- alge- Now let P = /, and therefore, is a primitive modular which Then a two-sided modular. part is true for any algebra. A is symmetric In this Two results 3.5, 3.6). if and only if it is maximal map of A onto A/R, then d(l) = l/R mutative -algebras. are symmetric. / be a primitive PC/. ideal in A/R. But A/R implies / = d " 1 ÍI/R) Banach *-algebra, License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use then com- is maximal Q.E.D. It is well known that if A is a commutative ideal If d is the quotient A is 334 WAYNETILLER symmetric if and only if every Theorem 5.2. is Hermitian // A is symmetric (hence Proof. Let MLF on A is Hermitian. and P-commutative, then every MLF on A positive). / be a MLF on A. Then R = ] Cket(f) since ker(/) tive ideal in A. If we define / ' : A/R —>C by / ' (x + P) = fix), Hermitian A/R MLF on A/R since is commutative. is a primi- then / ' is a Therefore, / is Hermitian. Q.E.D. The converse P-commutative of Theorem algebras Example 5.3. e, involution Let semisimple tions, Banach on D, then A be a noncommutative -algebra (x, y, z) Example with identity on D. thus + z Let z) which cannot We remark modular between two-sided ideals implies R = J; therefore as used for the analogous Theorem 5.4. PC/ Let /.-norm. But since Then in D. algebra D is just Thus, let and so A/I MLF's. of A. Set </>R is and commuta- It remains to show that y = - (x~ ) and z) = (0, 0¡ e D is not symmetric. then there is a one-to- on A and the set of maximal / be such an ideal. is commutative. result now applies and P-commutative. A be symmetric opera- the pro- no nonzero B is symmetric and symmetric, Indeed, commutative symmetric, Now if cb is a MLF A has element the set of all MLF's of A. identity and involution D pointwise cb is Hermitian. if A is P-commutative one correspondence by e also) (x, y, z) = (e + y x, e + x y, e + z be invertible that having by <7j[(x, y, z)] = </> (z), where x be an invertible (e, e, e) + (x, y, z) -algebra Give x, y £ A since tive, Then nonsymmetric ß be a commutative, (denoted 4>: D—> C, defined D is not symmetric. exist D is P-commutative. a MLF on B, are the only MLF's z £ B. Let = (y , x , z ), and 3.8; therefore, </>d's are Hermitian; is, there Banach MLF's. 4>[ix, y, 0)] = 0 for every the functionals all the that MLF is Hermitian. D = i(x, y, z): x, y £ A, z £ B\. involution of B with Thus, every x—» x , and no nonzero z —>z . Now set duct 5.2 is not true; on which Then A symmetric The (see same argument [6, p. 192]). Then R - ¡x £ A: vix) = 0|. Proof. x £ A, vix) v (x) and the theorem implies Since = 0. is clear). 5.5. Then ker(/) that x e P implies is P-commutative = 0; hence x e (I Theorem 4.3). Lemma A R . = P , we may assume vAiR(x) Therefore, P = /, it follows Since / that (usual 0= /'(x A has an identity. definition) + P) = fix) (f £ M) implies v(x) and symmetric, = 0, Conversely, and since Let f £ M (if M = 0. is a MLF on A/R. since A/R But vix) is commutative. x £ R (see the paragraph following Q.E.D. Suppose A is symmetric, P-commutative, License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use and has let vA (x) = an identity. = 0 P-COMMUTATIVE BANACH *-ALGEBRAS Then for every Proof. x in A, a(x) = {fix): The containment 4.3 and [4, Theorem (x + P) = fixfa A, it follows the lemma follows. Q.E.D. Theorem 5.6. Let < vix)v(y) Proof., < (sup over all give Example assume that Let Then Theorem x and without for every x, y £ 5.5, v(xy) the supremums = ate taken + sup|/(y)| = algebra seminorm in a the assumption of symmetry, note that extended P. This Example that -algebra algebra 5.3). We may A of Example It is easy vA [(x, y)iy, to check x)\ = vA [(xy, yx)] with an approximate identity. If = il Ker(/), is proved where situation- Now let since is over all = fiyx) for every and every positive x £ A, |/(x)| x, y £ A; A is therefore with an iden- RA = P / be an arbitrary for every 3] fixy) the intersection in [6, p. 259] for algebras < /' x, y e A. functional (e) vix x) Hence P-commutative. xy- Q.E.D. BIBLIOGRAPHY 1. J. W. Baker and J. S. Pym, A remark on continuous bilinear mappings, Proc. Edinburgh Math. Soc. 17 (1971), 245-248. 2'. J. Dixmier, Les C -algebres et leurs représentations, 2ieme ed., Cahiers Scienti- fiques, fase. 29, Gauthier-Villars, Paris, 1969. MR 39 #7442. 3. J. W. M. Ford, 42(1967), 521-522. A square = A is P-commutative. to A . Then = RA for every with an identity ^y¡ *^- to the present and by [l, Corollary -algebra P-commutative (see It follows is representable. / ' its extension Banach v„ (xy) > v„ (x) vR (y). Now form the x £ A, then on A. functional < / (e)i/(x), by Lemma where is a continuous that, A be a Banach for every We first yx e MKer(/) Then + y)| < sup|/(x)| y satisfying va ^x- y^va Let functionals on A and v(-) A is not symmetric 5.8. It is easily positive Then viy), ß be a noncommutative vb ^= x) < vix) tity. oA/fR(x + R) Co(x), -algebra. to show v„ (y) = v„ (x). B. Proof. f £ M. Since seminorm. oA L(x, y)] = oß (x) U oß iy). positive Since P-commutative. = vix) Banach an example 5.7. vb ^xy^> v b ^ vix for some + R). an identity. 5.6 we see that two elements that 3.8 using and v(x + y) = sup|/(x not be an algebra and having r\£ oA/R(x (sup|/(y)|) P-commutative need A= fix) from Theorem Q.E.D. We next v(-) A has |/(x)|) From Theorem symmetric, that immediately v(x + y) < v (x) + viy). / in M. Similarly, vix) + viy). that A be symmetric and We may assume sup|/(xy)| f £ M\ follows Now suppose / A, v(xy) f £ M\. o~(x) C \fíx): 4.10]. 335 root lemma for Banach *-algebras, MR 35 #5950. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use J. London Math. Soc. 336 WAYNE TILLER 4. J. W. M. Ford, Dissertation, Subalgebras Newcastle 5. I. S. Murphy, upon of Banach Tyne, Continuity algebras generated by semigroups, Ph.D. 1966. of positive linear functionals on Banach -algebras, Bull. London Math. Soc. 1 (1969), 171- 173. MR 40 #3320. 6. M. A. Naimark, Noordhoff, 7. C. E. Rickart, Van Nostrand, 9. General Princeton, 8. S. Shirali, 145-150. Normed rings, ''Nauka", Moscow, 1968; English theory of Banach algebras, University transi., Wolters- 1970. N.J., I960. Representability Series in Higher Math., MR 22 #5903. of positive functionals, J. London Math. Soc. (2) 3 (1971), MR 43 #2517. N. Th. Varopoulos, Continuité des formes linéaires positives sur une algebre de Banach avec involution, C. R. Acad. Sei. Paris 258 (1964), 1121- 1124. MR 28 #4387. DEPARTMENT OF MATHEMATICS, TEXAS CHRISTIAN UNIVERSITY, FORT WORTH, TEXAS 76109 Current address: Bocholt Marienschule, 429 Bocholt, License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use Schleusenwall 1, West Germany

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