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∗-Semisimplicity of tensor product LMC ∗-algebras
JOURNAL
OF MATHEMATICAL
*-Semisimplicity
ANALYSIS
AND
90,
APPLICATIONS
of Tensor
Product
43 l-439
(1982)
LMC *-Algebras1
MARIA FRAGOULOPOULOU Mathematical
Institute,
Unicersity Submitted
of Athens,
Athens
143. Greece
by K. Fan
Each (Hausdorff) Imc C*-algebra is *-semisimple. The *-semisimplicity of two suitable Imc*-algebras is passed on to their completed F-tensor product iff F is faithful. A sort of strong converse is also valid. In the commutative case, *semisimplicity implies semisimplicity, whereas the converse occurs for suitable Imwalgebras.
1. INTRODUCTION This paper deals with semisimplicity and, in particular, *-semisimplicity of topological tensor product algebras. Thus, Section 2 contains, the necessary background material. In Section 3 we give conditions under which semisimplicity of topological algebras is passed on to their tensor product (Theorem 3.1). The results obtained extend previous ones due to Gil de Lamadrid (71 within the Banach algebras theory, as well as to Smith [22] and Mallios [ 161 within the more general context of topological algebras. Furthermore, considering the tensorial lmc C*-topologies 01, v 161, which are the analogues of the two standard least and greatest tensorial C*-norms, we prove that a is faithful (Proposition 3.3), whereas this is not valid for v (Remark 3.3). In Section 4 we deal with *-semisimplicity, extending various Banach *algebra results mainly due to Laursen [ 111. Thus, Proposition 4.1 characterizes +-semisimplicity of an lmc*-algebra E in terms of the natural imbedding of E into its enveloping algebra. Besides, by the faithfulness of u, Theorem 4.2 shows that *-semisimplicity for lmc *-algebras is a hereditary property for K-tensor products iff d is faithful. A strong converse to the latter is also proved (Theorem 4.4). As a consequence, *-semisimplicity of an appropriate lmc *-algebra E is passed on to E-valued group and function algebras (Corollaries 4.2, 4.3). In the last section the previous kinds of semisimplicity, showing that *+ This paper relies on a part of the author’s Ph. D. Thesis, University of
Athens.
431 0022.247X;82/ 409.‘90
2 I I
12043 I-o9SO2.00;0
Copyright 0 1982 by Academic Press, Inc. All rights of reproduction in any form rescrvcd.
432
MARIAFRAGOULOPOULOU
semisimplicity implies semisimplicity in the commutative case, are related. The converse holds also true for algebras, which are essentially of the form C,(X), with X a completely regular space (cf. proof of Theorem 5.1).
2. PRELIMINARIES
The vector spaces and algebras considered throughout are over the field C of complex numbers, whereas the topological spaces involved are supposed to be Hausdorff. For the notation and terminology applied we refer to [5,6] a natural continuation of which the present paper constitutes. By an lmc *-algebra we mean a *-algebra E, endowed with a topology provided by a family of *-preserving submultiplicative seminorms, say (p,), a E A [ 18, 191. If in addition, p,(x*x) =pa(x)*, x E E, a E A, E is called an lmc C*-algebra (cf., for instance, [9]). A bai (bounded approximate identity) of E is a net (ei)ic, in E, with P,(e,)< 1, aEA, iEZ and lime,x=x= limxe,, x E E. Furthermore, a representation of an lmc *-algebra E is a *-morphism 4 from E into the C*-algebra Y(H,) of all bounded linear operators on some Hilbert space H,. Thus, R(E)(resp. R’(E)) denotes the set of all continuous (resp. continuous topologically irreducible) representations of E, while 9(E) stands for the continuous positive linear forms on E and 9(E) for the nonzero extreme points of the latter (cf. also [5]). On the other hand, if E has also a bai B(E) will denote the enveloping algebra of E [5, Def. 4.11, that is the Hausdorff completion of (E, (r,)) with r,(x) = sup{]ld(x)]] : ( E R;(E)}, x E E where R:(E) = {$ E R’(E): II$(x)ll
(iii) If M, N are equivontinuous subsets of Ej(weak topological dual of E), F: respectively, then M @ N is an equicontinuous subset of (E 0 F):. In this regard, if 71, E are the projective and biprojective tensorial
TENSORPRODUCTLMC
*-ALGEBRAS
433
topologies, respectively, [8, 121, then 8 < d < 7c,i.e., K is compatible in the sense of [ 12, Def. 3.11. Besides, 7~is always an admissible topology and the same holds true also for E in case it makes a topological tensor product into an lmc *-algebra (cf. [6, Remark 2.11). On the other hand, given the complete locally convex spaces E, F a compatible topology d on E @ F is calledfaithful [ 13, Def. 2.11, if the map i,:EaF4@FF2e(Ej,F:)
F
E
(2-l)
with i&)(x’, y’) = (x’ @y’)(z) for any z E E 8, F, and (x’, y’) E El x F: is l-l, where 2,,(Ej, F:) is the space of separately continuous bilinear forms on Ej X Fb, endowed with the topology of biequicontinuous convergence.
3. SEMISIMPLICITY A topological algebra E is said to be semisimpleif the respective Gel’fand map is l-l, equivalently, if 0 {kerCf):fE 9JI(E)} = {O}, where YJI(E) denotes the spectrum (Gel’fand space) of E (cf., for instance, [ 16, p. 121 and/or 1181).
Every semisimple topological algebra in the preceding sense is, of course, commutative, this being a consequence of the functional character of the definition, different from the one we usually have (cf., for example, [ 21, Def. 2.3.1; 17, p. 474, Scholium]). In this respect, the next result constitutes a broad extension of 122, Theorem 4 1. THEOREM 3.1. Let E, F be semisimpletopological algebras with locally equicontinuous spectra and 8 a compatible topology on E 0 F in the senseof [ 12, DeJ 3.11, in such a way that fm(E OS F) = 9JI(E 6,, F), within a bijection. Then, E 6, F is semisimpleiff E’ @ F’ = (E 6, F); .
Proof. (Only if). By hypoth esis and [18, Lemma IV, 3.11, (E GCF): = [9JI(E Og F)] (closed linear hull of IDI(E & F)) (cf. also [22, Remark 3]), so that if w E (E GK F);, w = lim,(Z,A,h,),, with [ 15, Theorem 2.11
h, =f,, 0 g, E m(E &d’), df,, g,) E mm(E) x ‘Wf’). (If). For 0 # z E E g)d F, there exists by hypothesis z’ =x’ 0 y’ E with (x’, y’) E E’ x F’, such that (x’ @y’)(z) # 0, i.e., (E @A’, ~‘(0, ,(z)) f 0, where &, is the continuous extension of #,, : E OS F + F [ 16, Lemma 5.11. But then, 0 + &(z) E F, so that, since F is semisimple there exists g E 9R(F), with 0 f g(&Jz)) = x’(F~(z)), hence 0 # JJz) E E, which by the semisimplicity of E yields f(Jg(z)) # 0, for some fE YJI(E). Thus, df@ g)(z) # 0, withf@ g E !IJI(E @)dF)[ 15, Theorem 2.11. 1
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MARIA FRAGOULOPOULOU
In case E, F are locally convex, Theorem 3.1 is still equivalent to the fact that d is a faithful topology according to [ 13, Theorem 2.21 and [ 15, Sect. 3, 11. In this regard, the result has also to be compared with [8, I, Proposition 3.7(a) o (b)]. On the other hand, with E, F, & as in Theorem 3.1, let f?,(E;, F;) be the set of all separately continuous bilinear forms on Ei x F,’ with a vector space topology K not smaller than the topology of simple convergence, and i, the continuous canonical linear map of E gKF into f?!,(E:, F;) such that i,(z)(x’,y’) = (x’ @v’)(z) f or any (x’, y’) E Ei X Fi, z E E 6, F. Then, Theorem 3.1 is also equivalent to the injectivness of i, as this follows by [ 16, Theorem 5.31 and [ 15, Sect. 3, 11. Now, let E, F be two Imc C*-algebras. In [6] we have defined the tensorial Imc C*-topologies a, v, analogues of the standard least and greatest tensorial C*-norms, respectively. The submultiplicative seminorms for a, v are given by t,,&)
= supO](# 0 y/)(z)]1 : ((3 w) E R,(E) x ZW’)], zEE@Fy
7
zEE@F,
(a,&EA
XB
(a,/3)EA
xZ3,
respectively. Moreover, a, v are Hausdorff admissible topologies on E @ F with a < v and a = v, when either of E, F is of type Z [6]. Regarding a we still have PROPOSITION 3.3. Then, a isfaithful.
Let E, F be complete barrelled Q Imc C*-algebras.
Proof: By [ 10, Theorem 31 the respective unital algebras E,, F, of E, F [5] are identified with C*-subalgebras of Y(ZZ,), Y(H,) for some Hilbert spaces H, , H,, so that the same holds also true for E, F (cf. [6, proof of Theorem 4.1 I). Thus, E $, F becomes a C*-subalgebra of Y(H, G,, H,) [6, proof of Theorem 4.11, where u is the canonical cross norm on H, @ H,. This is what we actually need for the injectivness of i,, the proof of which being, however, an easy adaptation to our case of the respective proof of [ 11, Lemma 6.61. I Remark 3.3. Proposition 3.3 is not valid for v. Since a < v, i,, :E 0, F + E 0, F is continuous, so that it is continuously extended (denoted by the same symbol) to E 6, F, with i, o i,, = i,. Thus, if v is faithful, i,, is l-l and since E @,, F, E G,, F are in fact C*-algebras (cf. proof of Proposition 3.3, as well as [6, proof of Theorem 4.1 I), i,, has a continuous
inverse, that is, one finally gets a = v which is, however, not true in general (231 (cf. also [ 11, Remark 6.71).
TENSOR
PRODUCT
4.
LMC
*-ALGEBRAS
435
*-SEMISIMPLICITY
Let E be a topological *-algebra. Then, R *(E) = (-I (ker(4) : $ E R’(E)} is called the *-radical of E, and if R*(E) = (0}, E is said to be *-semisimple. PROPOSITION 4.1. Let E be an lmc *-algebra with a bai and B(E) its enveloping algebra. Then, E is *-semisimple iff the canonical (enveloping) map rE: E + B(E) is l-l.
Proof. Let x E E with rE(x) = 0. Then, if (qa)aeA is the family of submultiplicative seminorms defining the topology of B(E), we have q,(x + I) = 0 for every a E A, therefore 4(x) = 0 for every $ E R:(E) and (x E A, hence x = 0. Conversely, if x E R*(E), b(x) = 0 for each 4 E R’(E), so that r,(x) = 0 for every a E A, that is, x E I; therefore, TV = 0 and hence x = 0. !
An immediate consequence of Proposition 4.1 is that every lmc C*-algebra Besides, such an algebra always has a bai [9, Theorem 2.6). Regarding *-semisimplicity of a topological tensor product E GK F, with E, F lmc *-algebras and d an admissible topology on E @ F, we prove (Theorem 4.2) that this depends on the *-semisimplicity of E, F and the faithfulness of @F.A sort of a strong converse is also valid (Theorem 4.4). Before we proceed to Theorem 4.2, we make some more comments on the terminology applied. Thus, suppose E, F are locally convex spaces and Y(El, F) (resp. 5f(F:, E)) the continuous linear maps of E: (res. F:) into F (resp. E). Then, if d is a faithful topology on E G)d F, the map E 6,, F + rP(Ej , F) (resp. 9(F;, E)) : L I-+ T, (resp. S,) with T,(x’) = JXS(z), x’ E El (for TX, cf. proof of Theorem 3.1 as well as [ 16, Lemma 5.11) is certainly injective. Besides, let ‘TV : Z’(E)’ + E’ be the transpose of rE (Proposition 4.1). such that %,(a’) =f,, , a’ E B(E)’ with f,,(x) = (a’ o rE)(x), x E E. In this concern, our hypothesis in the next theorem, one of the algebras to be of type I, is actually made for ensuring a = v (cf. 16, relation (3.5)]), so that the same result could be stated in this strengthened form as well. The latter constitutes, in fact, the extension in our case of a Banach *-algebra analogue in [ 11, Theorem 6.91. is *-semisimple.
THEOREM
4.2.
Let E, F be complete bQ
lmc
*-semisimple algebras with
bai s and Er an admissible topology on E @ F in such a way that st. 0 rp: E OFF + B(E) @,, B(F) is contiinuous. Also, let either E or F be of type I. Then. E @ F is *-semisimple iff & is faithful. Proof. Suppose E &)d F is *-semisimple. If z E E aB F with i,(z) = 0. then (x’ @y’)(z) = 0 for any x’ E E’, 4” E F’, so that (f,, @fb,)(z) = 0 for any a’ E d(E)‘. b’ E a(F)‘. Thus, if j, is the map (2.1) corresponding
436
MARIA FRAGOULOPOULOU
to B(E) &,, B(F), one obtains 0 = ((a’ o rE) @ (6’ o rF))(z) = ((a’ 0 6’) 0 (rE 0 rF))(z) = j,((r, @ r,)(z))(u’, b’) for any a’ E F(E)‘, b’ E 8(F)‘. Hence,j,((r, @ rF)(z)) = 0, that is, (j, 0 ,U o r)(z) = 0, where ,U is the topological algebraic isomorphism between 8(E &)d F) and 8(E)-& B(F) (cf. [6, Corollary 4.1, proof of Theorem 4.11) and r : E OS F + 8(E &F) the respective canonical enveloping map. But, each one of the j,, ,u, r is l-1 (cf. Proposition 3.3, [6, Corollary 4.11 and Proposition 4.1 respectively), consequently z = 0. Conversely, by Proposition 4.1, it suffices to show that r is injective. Thus, if O#z,E E &-I;, is(z,)=z#O, so that T, #O (cf. discussion before Theorem 4.2). Hence, there is x’ E E’ with 0 # T&c’) E F, so that 0 # rF(Tz(x’)) E B(F) (Proposition 4. I), which in turn implies b’(r,(T,(x’))) # 0 for some b’ E Z’(F)‘. But, b’(r,(T,(x’))) = x’(S,(‘r,(b’))), therefore 0 # S,(&,) E E, hence (Proposition 4.1) 0 # rE(Sz(fbt)) E B(E). Thus, there is a’ E B(E)‘, with u’(rE(.!3,dfbr))) # 0, and since (r, 0 rF)(z,,) # 0, that is, a’MUL))> = (a’ 0 b’WE 0 Qh)>, (,u o r)(z,,) # 0 (cf. [6, proof of Theorem 4. l]), which yields r(zo) f 0. 1 COROLLARY 4.2. Let X be a compact space and E a complete bQ Imc *algebra with a bai. Then, C,(X, E)(E-valued continuous functions on X endowed with the topology of unijbrm convergence in X) is *-semisimpleiff E is *-semisimple.
Proof: C,(X, E) = C,(E) @,, E, within algebraic a topological isomorphism [17, Lemma 4.11, where C,(X) is a type I [3j C*-algebra, hence *-semisimple (Proposition 4.1). On the other hand, E is an admissible faithful topology, which makes the map idcUcx,@ rE : C,(X) O,E -+ continuous (cf. [6, Remark 2.1 and proof of Corollary 4.3]), C,(x) 6% g(E) so that the assertion follows by Theorem 4.2. m
The reason for having X compact in Corollary 4.2 is that by Theorem 4.2 C,(X) should be a Q-algebra, which is, in fact, the case iff X is compact 12, Theorem 2.1.5 (i)J. COROLLARY 4.3. Let G be a locally compact group and E a complete bQ Imc *-algebra with a bai. Then, the generalized group algebra L;(G) of G is * -semisimplei ff E is * -semisimple.
Proof. LA(G) = L ‘(G) Q,, E within a topological algebraic isomorphism [ 141, where L’(G.) is a type I [3] *-semisimple 1211 Banach algebra with a bai, which besides, satisfies the condition of approximation [ 81. Hence, n is a faithful topology [ 131, so that the conclusion now follows by [6, Lemma 4.11 and Theorem 4.2. a
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LMC *-ALGEBRAS
To get a converse to Theorem 4.2, we next prove that R*(E) is the intersection of the kernels of all continuous representations of E. In this concern. if fE 9(E), L,zz (x E E $(x*x) = 0) = ker(#& where #f is the continuous representation of E corresponding to f by the GNS-construction (cf. [S. Theorem 3.41). On the other hand, if RE denotes the Hermitian part of E, an application of [5, Theorem 3. I, Corollary 3.1, Lemma 3.3; 121. Lemma 4.6.6 ] yields that For every y EXE with f(y) # 0 for some f E 9(E), g E 9(E) such that g(y) # 0.
there is (4.1)
Thus. we now come to the promised result. 4.4. Let E be an lmc *-algebra with a bai. R*(E) = n (ker(d) : 4 E R(E)}. PROPOSITION
Then.
Proof. If x E E, with d,,(x) # 0 for some &, E R(E), then &,(x*) # 0 too. Hence, there exists <,, E H,,,, with ]]&,]I < 1 and q$,(x*)(&,) # 0. so that if f(y) = (q&(y)(&), &,), y E E,f is an element of Y(E) with f (xx*) # 0. Now. by (4.1) there exists h E 9(E) with h(xx*) # 0, so that h(xx*xx*) f 0 by (5, Lemma 3.3 ii)], i.e., xx* @ L, = ker(#,), hence x 4 R*(E) (cf. also [S. Proposition 3.61). ti
Proposition 4.4 represents in our case the standard Banach *-algebras result [21, Theorem 4.6.71. The next theorem provides a sort of a stengthened converse to Theorem 4.2, which besides extends ] 11. Proposition 6. lo]. THEOREM 4.4. Let E, F be lmc *-algebras with bai s and a an admissible topology on E @ F. Suppose also that E @jj, F is *-semisimple. Then, E, F are *-semisimple.
Proof. If (x,yjE E x F withx#O,y#O, thenx@y& R*(EhKF), i.e., there is d E R ‘(E hK F) with 0(x By) # 0. Thus, if dE, QF are the restrictions of $ to E, F, respectively, [6, Lemma 3.31, one has )(x @ y) = dE(x) br(y), so dE(x) # 0. d,(y) # 0, which yields the assertion by Proposition 4.4. m
5. COMMUTATIVE
*-SEMISIMPLE
We relate below, in the case of commutative semisimplicity considered before.
ALGEBRAS
algebras, the two kinds of
LEMMA 5.1. Let E be a commutative Imc *-algebra with a bai. Then. .9(E) = 9(E) n YIN(E), within a bijection. In other words, .9(E) = 2’ *(E). with Q:*(E) the Hermitian spectrum of E.
438
MARIA FRAGOULOPOULOU
Proof: If E, is the respective unital algebra of E [5] and j?, its completion, one easily gets !Ui’(E) = W(E,) = %I@,) within bijections. Thus, the assertion follows by [l, Theorem 4.51 and [5, Proposition 3.41. 1 Remark 5.1. [4, Corollary 3.81 gives a Bochner-Weil-Raikov theorem for lmc *-algebras: that is, if E is a commutative lmc *-algebra with continuous Gel’fand map, in such a way that (x*)- = (X)-, x E E, then SEE’ is positive and extendable ifff(x) =,~(x^), x E E, with ,u a (complex) positive finite measure on !IR(E). (As a matter of fact we do not really need the continuity of the Gel’fand map). Now, if E has .a bai, everyfE 9(E) is extendable [5, Lemma 3.31, so that by restriction to g*(E), one gets the next abstract form of the above result:
Let E be a commutative lmc *-algebra with a bai. Also 1etfE E’. Then, f is positive iff f(x) = p(J), x E E, with p a positive finite measure on g*(E). Concerning the Banach *-algebras analogue of the latter (cf. [20, Corollary 9.141). In this regard, we still remark that Lemma 5.1 could also be derived from the preceding abstract form of Bochner-Weil-Raikov theorem. THEOREM 5.1. Let E be a commutative lmc *-algebra with a bai. Then, *-semisimplicity implies semisimplicity on E. Furthermore, the two notions coincide if E is, in particular, a commutative unital complete lmc C*-algebra with continuous Gel’fand map.
ProoJ
According to 120, Corollary 6.41 and Lemma 5.1 R’(E) = 2?*(E) =9(E)
E W(E),
so that n (ker(f), fE W(E)} c R*(E), which proves the first part of the assertion. Concerning the second part of the statement, the hypothesis implies that E = C,(!JJI(E))(the algebra of continuous complex-valued functions on W(E) with the topology of compact convergence) within a topological algebraic isomorphism (Gel’fand-Naimark theorem; cf. [ 19, Theorem 8.4; 18, Lemma IV, 3.21). Thus, setting X = 1132(E), X= %Jl(C,(X)) = G*(C,(X))
= R’(C,(X)),
within bijections the first being actually a homeomorphism 9.21. 1
[ 18, Theorem III,
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ACKNOWLEDGMENTS The author expresses her appreciation to Professor Anastasios Mallios for helpful and stimulating discussions during the preparation of this paper.
REFERENCES 1. R. M. BROOKS, On locally m-convex *-algebras, Pacific J. Math. 23 (1967), S-23. 2. W. J. DIEDRIECH, “Function Algebras on Completely Regular Spaces,” Ph. D. Disser~ tation, Northwestern University, Evanston, Illinois, 1971. 3. J. DIXMIER, “Les C*-Algtbres et leurs Reprksentations,” Gauthier-Villars, Paris, 1964. 4. M. FRAGOULOPOIJLOU,Integral representations of linear forms on topological algebras. Comment. Math. 21 (1979), 43-53. 5. M. FRAGOLJLOPOULOU,Spaces of representations and enveloping 1.m.c. *-algebras, Pacific J. Much. 95 (1981), 61-73. 6. M. FRAGOULOPOULOU,“Representations of tensor product 1.m.c. *-algebras,” to appear. 7. J. GIL DE LAMADRID, Uniform cross norms and tensor products of Banach algebras. Duke Math. J. 32 (1965), 359-368. 8. A. GROTHENDIECK, Produits tensoriels topologiques et espaces nuclkaires, Mem. Amer. Math. Sot. (16) (1955). 9. A. INOUE, Locally C*-algebra, Mem. Fat. Sci. Kyushu Univ. Ser. A 25 (1971). 197-235. 10. G. LASSNER, uber Realisirungen gewisser *-Algebren, Math. Nachr. 52 (1972), 161-166. I I. K. 8. LAURSEN, Tensor products of Banach algebras with involution, Truns. Amer. Math. Sot. 136 (1969), 467-487. 12. A. MALLIOS, On the spectrum of a topological tensor product of locally convex algebras, Math. Ann. 154 (1964), 171-180. 13. A. MALLIOS. Heredity of tensor products of topological algebras. Marh. Ann. 162 ( 1966 ). 246-257. 14. A. MALLIOS, Note on the tensor products and harmonic analysis, Mafh. Ann. 173 (1967). 287-289. 15. A. MALLIOS, Spectrum and boundary of topological tensor product algebras, Bull. Sot. Mafh. G&e 8 (N.S.) (1967), 101-I 15. 16. A. MALLIOS, Semisimplicity of tensor products of topological algebras. Bull. Sot. Mafh. G&e (N. S.) 8 (1967), 1-16. 17. A. MALLIOS. On functional representations of topological algebras, J. Puncr. Anal. 6 (1970). 468480. 18. A. MALLIOS, “General Theory of Topological Algebras. Selected Topics.” in preparation. 19. E. A. MICHAEL, Locally multiplicatively-convex topological algebras. Mem. Amer. Math. Sot. (11) (1952). 20. R. D. MOSAK, “Banach Algebras,” Univ. of Chicago Press. Chicago/London, 1975. 21. C. E. RICKART. “General Theory of Banach Algebras,” Van Nostrand. Princeton. N. J.. 1960. 22. H. A. SMITH. Tensor products of locally convex algebras. Proc. Amer. Math. Sot. 17 (1966). 124-132. 23. M. TAKESAKI. On the cross norm of the direct product of C*-algebras. Tohoku Mach. J. 16 (1964), 1 I I-122.
OF MATHEMATICAL
*-Semisimplicity
ANALYSIS
AND
90,
APPLICATIONS
of Tensor
Product
43 l-439
(1982)
LMC *-Algebras1
MARIA FRAGOULOPOULOU Mathematical
Institute,
Unicersity Submitted
of Athens,
Athens
143. Greece
by K. Fan
Each (Hausdorff) Imc C*-algebra is *-semisimple. The *-semisimplicity of two suitable Imc*-algebras is passed on to their completed F-tensor product iff F is faithful. A sort of strong converse is also valid. In the commutative case, *semisimplicity implies semisimplicity, whereas the converse occurs for suitable Imwalgebras.
1. INTRODUCTION This paper deals with semisimplicity and, in particular, *-semisimplicity of topological tensor product algebras. Thus, Section 2 contains, the necessary background material. In Section 3 we give conditions under which semisimplicity of topological algebras is passed on to their tensor product (Theorem 3.1). The results obtained extend previous ones due to Gil de Lamadrid (71 within the Banach algebras theory, as well as to Smith [22] and Mallios [ 161 within the more general context of topological algebras. Furthermore, considering the tensorial lmc C*-topologies 01, v 161, which are the analogues of the two standard least and greatest tensorial C*-norms, we prove that a is faithful (Proposition 3.3), whereas this is not valid for v (Remark 3.3). In Section 4 we deal with *-semisimplicity, extending various Banach *algebra results mainly due to Laursen [ 111. Thus, Proposition 4.1 characterizes +-semisimplicity of an lmc*-algebra E in terms of the natural imbedding of E into its enveloping algebra. Besides, by the faithfulness of u, Theorem 4.2 shows that *-semisimplicity for lmc *-algebras is a hereditary property for K-tensor products iff d is faithful. A strong converse to the latter is also proved (Theorem 4.4). As a consequence, *-semisimplicity of an appropriate lmc *-algebra E is passed on to E-valued group and function algebras (Corollaries 4.2, 4.3). In the last section the previous kinds of semisimplicity, showing that *+ This paper relies on a part of the author’s Ph. D. Thesis, University of
Athens.
431 0022.247X;82/ 409.‘90
2 I I
12043 I-o9SO2.00;0
Copyright 0 1982 by Academic Press, Inc. All rights of reproduction in any form rescrvcd.
432
MARIAFRAGOULOPOULOU
semisimplicity implies semisimplicity in the commutative case, are related. The converse holds also true for algebras, which are essentially of the form C,(X), with X a completely regular space (cf. proof of Theorem 5.1).
2. PRELIMINARIES
The vector spaces and algebras considered throughout are over the field C of complex numbers, whereas the topological spaces involved are supposed to be Hausdorff. For the notation and terminology applied we refer to [5,6] a natural continuation of which the present paper constitutes. By an lmc *-algebra we mean a *-algebra E, endowed with a topology provided by a family of *-preserving submultiplicative seminorms, say (p,), a E A [ 18, 191. If in addition, p,(x*x) =pa(x)*, x E E, a E A, E is called an lmc C*-algebra (cf., for instance, [9]). A bai (bounded approximate identity) of E is a net (ei)ic, in E, with P,(e,)< 1, aEA, iEZ and lime,x=x= limxe,, x E E. Furthermore, a representation of an lmc *-algebra E is a *-morphism 4 from E into the C*-algebra Y(H,) of all bounded linear operators on some Hilbert space H,. Thus, R(E)(resp. R’(E)) denotes the set of all continuous (resp. continuous topologically irreducible) representations of E, while 9(E) stands for the continuous positive linear forms on E and 9(E) for the nonzero extreme points of the latter (cf. also [5]). On the other hand, if E has also a bai B(E) will denote the enveloping algebra of E [5, Def. 4.11, that is the Hausdorff completion of (E, (r,)) with r,(x) = sup{]ld(x)]] : ( E R;(E)}, x E E where R:(E) = {$ E R’(E): II$(x)ll
(iii) If M, N are equivontinuous subsets of Ej(weak topological dual of E), F: respectively, then M @ N is an equicontinuous subset of (E 0 F):. In this regard, if 71, E are the projective and biprojective tensorial
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topologies, respectively, [8, 121, then 8 < d < 7c,i.e., K is compatible in the sense of [ 12, Def. 3.11. Besides, 7~is always an admissible topology and the same holds true also for E in case it makes a topological tensor product into an lmc *-algebra (cf. [6, Remark 2.11). On the other hand, given the complete locally convex spaces E, F a compatible topology d on E @ F is calledfaithful [ 13, Def. 2.11, if the map i,:EaF4@FF2e(Ej,F:)
F
E
(2-l)
with i&)(x’, y’) = (x’ @y’)(z) for any z E E 8, F, and (x’, y’) E El x F: is l-l, where 2,,(Ej, F:) is the space of separately continuous bilinear forms on Ej X Fb, endowed with the topology of biequicontinuous convergence.
3. SEMISIMPLICITY A topological algebra E is said to be semisimpleif the respective Gel’fand map is l-l, equivalently, if 0 {kerCf):fE 9JI(E)} = {O}, where YJI(E) denotes the spectrum (Gel’fand space) of E (cf., for instance, [ 16, p. 121 and/or 1181).
Every semisimple topological algebra in the preceding sense is, of course, commutative, this being a consequence of the functional character of the definition, different from the one we usually have (cf., for example, [ 21, Def. 2.3.1; 17, p. 474, Scholium]). In this respect, the next result constitutes a broad extension of 122, Theorem 4 1. THEOREM 3.1. Let E, F be semisimpletopological algebras with locally equicontinuous spectra and 8 a compatible topology on E 0 F in the senseof [ 12, DeJ 3.11, in such a way that fm(E OS F) = 9JI(E 6,, F), within a bijection. Then, E 6, F is semisimpleiff E’ @ F’ = (E 6, F); .
Proof. (Only if). By hypoth esis and [18, Lemma IV, 3.11, (E GCF): = [9JI(E Og F)] (closed linear hull of IDI(E & F)) (cf. also [22, Remark 3]), so that if w E (E GK F);, w = lim,(Z,A,h,),, with [ 15, Theorem 2.11
h, =f,, 0 g, E m(E &d’), df,, g,) E mm(E) x ‘Wf’). (If). For 0 # z E E g)d F, there exists by hypothesis z’ =x’ 0 y’ E with (x’, y’) E E’ x F’, such that (x’ @y’)(z) # 0, i.e., (E @A’, ~‘(0, ,(z)) f 0, where &, is the continuous extension of #,, : E OS F + F [ 16, Lemma 5.11. But then, 0 + &(z) E F, so that, since F is semisimple there exists g E 9R(F), with 0 f g(&Jz)) = x’(F~(z)), hence 0 # JJz) E E, which by the semisimplicity of E yields f(Jg(z)) # 0, for some fE YJI(E). Thus, df@ g)(z) # 0, withf@ g E !IJI(E @)dF)[ 15, Theorem 2.11. 1
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In case E, F are locally convex, Theorem 3.1 is still equivalent to the fact that d is a faithful topology according to [ 13, Theorem 2.21 and [ 15, Sect. 3, 11. In this regard, the result has also to be compared with [8, I, Proposition 3.7(a) o (b)]. On the other hand, with E, F, & as in Theorem 3.1, let f?,(E;, F;) be the set of all separately continuous bilinear forms on Ei x F,’ with a vector space topology K not smaller than the topology of simple convergence, and i, the continuous canonical linear map of E gKF into f?!,(E:, F;) such that i,(z)(x’,y’) = (x’ @v’)(z) f or any (x’, y’) E Ei X Fi, z E E 6, F. Then, Theorem 3.1 is also equivalent to the injectivness of i, as this follows by [ 16, Theorem 5.31 and [ 15, Sect. 3, 11. Now, let E, F be two Imc C*-algebras. In [6] we have defined the tensorial Imc C*-topologies a, v, analogues of the standard least and greatest tensorial C*-norms, respectively. The submultiplicative seminorms for a, v are given by t,,&)
= supO](# 0 y/)(z)]1 : ((3 w) E R,(E) x ZW’)], zEE@Fy
7
zEE@F,
(a,&EA
XB
(a,/3)EA
xZ3,
respectively. Moreover, a, v are Hausdorff admissible topologies on E @ F with a < v and a = v, when either of E, F is of type Z [6]. Regarding a we still have PROPOSITION 3.3. Then, a isfaithful.
Let E, F be complete barrelled Q Imc C*-algebras.
Proof: By [ 10, Theorem 31 the respective unital algebras E,, F, of E, F [5] are identified with C*-subalgebras of Y(ZZ,), Y(H,) for some Hilbert spaces H, , H,, so that the same holds also true for E, F (cf. [6, proof of Theorem 4.1 I). Thus, E $, F becomes a C*-subalgebra of Y(H, G,, H,) [6, proof of Theorem 4.11, where u is the canonical cross norm on H, @ H,. This is what we actually need for the injectivness of i,, the proof of which being, however, an easy adaptation to our case of the respective proof of [ 11, Lemma 6.61. I Remark 3.3. Proposition 3.3 is not valid for v. Since a < v, i,, :E 0, F + E 0, F is continuous, so that it is continuously extended (denoted by the same symbol) to E 6, F, with i, o i,, = i,. Thus, if v is faithful, i,, is l-l and since E @,, F, E G,, F are in fact C*-algebras (cf. proof of Proposition 3.3, as well as [6, proof of Theorem 4.1 I), i,, has a continuous
inverse, that is, one finally gets a = v which is, however, not true in general (231 (cf. also [ 11, Remark 6.71).
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*-SEMISIMPLICITY
Let E be a topological *-algebra. Then, R *(E) = (-I (ker(4) : $ E R’(E)} is called the *-radical of E, and if R*(E) = (0}, E is said to be *-semisimple. PROPOSITION 4.1. Let E be an lmc *-algebra with a bai and B(E) its enveloping algebra. Then, E is *-semisimple iff the canonical (enveloping) map rE: E + B(E) is l-l.
Proof. Let x E E with rE(x) = 0. Then, if (qa)aeA is the family of submultiplicative seminorms defining the topology of B(E), we have q,(x + I) = 0 for every a E A, therefore 4(x) = 0 for every $ E R:(E) and (x E A, hence x = 0. Conversely, if x E R*(E), b(x) = 0 for each 4 E R’(E), so that r,(x) = 0 for every a E A, that is, x E I; therefore, TV = 0 and hence x = 0. !
An immediate consequence of Proposition 4.1 is that every lmc C*-algebra Besides, such an algebra always has a bai [9, Theorem 2.6). Regarding *-semisimplicity of a topological tensor product E GK F, with E, F lmc *-algebras and d an admissible topology on E @ F, we prove (Theorem 4.2) that this depends on the *-semisimplicity of E, F and the faithfulness of @F.A sort of a strong converse is also valid (Theorem 4.4). Before we proceed to Theorem 4.2, we make some more comments on the terminology applied. Thus, suppose E, F are locally convex spaces and Y(El, F) (resp. 5f(F:, E)) the continuous linear maps of E: (res. F:) into F (resp. E). Then, if d is a faithful topology on E G)d F, the map E 6,, F + rP(Ej , F) (resp. 9(F;, E)) : L I-+ T, (resp. S,) with T,(x’) = JXS(z), x’ E El (for TX, cf. proof of Theorem 3.1 as well as [ 16, Lemma 5.11) is certainly injective. Besides, let ‘TV : Z’(E)’ + E’ be the transpose of rE (Proposition 4.1). such that %,(a’) =f,, , a’ E B(E)’ with f,,(x) = (a’ o rE)(x), x E E. In this concern, our hypothesis in the next theorem, one of the algebras to be of type I, is actually made for ensuring a = v (cf. 16, relation (3.5)]), so that the same result could be stated in this strengthened form as well. The latter constitutes, in fact, the extension in our case of a Banach *-algebra analogue in [ 11, Theorem 6.91. is *-semisimple.
THEOREM
4.2.
Let E, F be complete bQ
lmc
*-semisimple algebras with
bai s and Er an admissible topology on E @ F in such a way that st. 0 rp: E OFF + B(E) @,, B(F) is contiinuous. Also, let either E or F be of type I. Then. E @ F is *-semisimple iff & is faithful. Proof. Suppose E &)d F is *-semisimple. If z E E aB F with i,(z) = 0. then (x’ @y’)(z) = 0 for any x’ E E’, 4” E F’, so that (f,, @fb,)(z) = 0 for any a’ E d(E)‘. b’ E a(F)‘. Thus, if j, is the map (2.1) corresponding
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to B(E) &,, B(F), one obtains 0 = ((a’ o rE) @ (6’ o rF))(z) = ((a’ 0 6’) 0 (rE 0 rF))(z) = j,((r, @ r,)(z))(u’, b’) for any a’ E F(E)‘, b’ E 8(F)‘. Hence,j,((r, @ rF)(z)) = 0, that is, (j, 0 ,U o r)(z) = 0, where ,U is the topological algebraic isomorphism between 8(E &)d F) and 8(E)-& B(F) (cf. [6, Corollary 4.1, proof of Theorem 4.11) and r : E OS F + 8(E &F) the respective canonical enveloping map. But, each one of the j,, ,u, r is l-1 (cf. Proposition 3.3, [6, Corollary 4.11 and Proposition 4.1 respectively), consequently z = 0. Conversely, by Proposition 4.1, it suffices to show that r is injective. Thus, if O#z,E E &-I;, is(z,)=z#O, so that T, #O (cf. discussion before Theorem 4.2). Hence, there is x’ E E’ with 0 # T&c’) E F, so that 0 # rF(Tz(x’)) E B(F) (Proposition 4. I), which in turn implies b’(r,(T,(x’))) # 0 for some b’ E Z’(F)‘. But, b’(r,(T,(x’))) = x’(S,(‘r,(b’))), therefore 0 # S,(&,) E E, hence (Proposition 4.1) 0 # rE(Sz(fbt)) E B(E). Thus, there is a’ E B(E)‘, with u’(rE(.!3,dfbr))) # 0, and since (r, 0 rF)(z,,) # 0, that is, a’MUL))> = (a’ 0 b’WE 0 Qh)>, (,u o r)(z,,) # 0 (cf. [6, proof of Theorem 4. l]), which yields r(zo) f 0. 1 COROLLARY 4.2. Let X be a compact space and E a complete bQ Imc *algebra with a bai. Then, C,(X, E)(E-valued continuous functions on X endowed with the topology of unijbrm convergence in X) is *-semisimpleiff E is *-semisimple.
Proof: C,(X, E) = C,(E) @,, E, within algebraic a topological isomorphism [17, Lemma 4.11, where C,(X) is a type I [3j C*-algebra, hence *-semisimple (Proposition 4.1). On the other hand, E is an admissible faithful topology, which makes the map idcUcx,@ rE : C,(X) O,E -+ continuous (cf. [6, Remark 2.1 and proof of Corollary 4.3]), C,(x) 6% g(E) so that the assertion follows by Theorem 4.2. m
The reason for having X compact in Corollary 4.2 is that by Theorem 4.2 C,(X) should be a Q-algebra, which is, in fact, the case iff X is compact 12, Theorem 2.1.5 (i)J. COROLLARY 4.3. Let G be a locally compact group and E a complete bQ Imc *-algebra with a bai. Then, the generalized group algebra L;(G) of G is * -semisimplei ff E is * -semisimple.
Proof. LA(G) = L ‘(G) Q,, E within a topological algebraic isomorphism [ 141, where L’(G.) is a type I [3] *-semisimple 1211 Banach algebra with a bai, which besides, satisfies the condition of approximation [ 81. Hence, n is a faithful topology [ 131, so that the conclusion now follows by [6, Lemma 4.11 and Theorem 4.2. a
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To get a converse to Theorem 4.2, we next prove that R*(E) is the intersection of the kernels of all continuous representations of E. In this concern. if fE 9(E), L,zz (x E E $(x*x) = 0) = ker(#& where #f is the continuous representation of E corresponding to f by the GNS-construction (cf. [S. Theorem 3.41). On the other hand, if RE denotes the Hermitian part of E, an application of [5, Theorem 3. I, Corollary 3.1, Lemma 3.3; 121. Lemma 4.6.6 ] yields that For every y EXE with f(y) # 0 for some f E 9(E), g E 9(E) such that g(y) # 0.
there is (4.1)
Thus. we now come to the promised result. 4.4. Let E be an lmc *-algebra with a bai. R*(E) = n (ker(d) : 4 E R(E)}. PROPOSITION
Then.
Proof. If x E E, with d,,(x) # 0 for some &, E R(E), then &,(x*) # 0 too. Hence, there exists <,, E H,,,, with ]]&,]I < 1 and q$,(x*)(&,) # 0. so that if f(y) = (q&(y)(&), &,), y E E,f is an element of Y(E) with f (xx*) # 0. Now. by (4.1) there exists h E 9(E) with h(xx*) # 0, so that h(xx*xx*) f 0 by (5, Lemma 3.3 ii)], i.e., xx* @ L, = ker(#,), hence x 4 R*(E) (cf. also [S. Proposition 3.61). ti
Proposition 4.4 represents in our case the standard Banach *-algebras result [21, Theorem 4.6.71. The next theorem provides a sort of a stengthened converse to Theorem 4.2, which besides extends ] 11. Proposition 6. lo]. THEOREM 4.4. Let E, F be lmc *-algebras with bai s and a an admissible topology on E @ F. Suppose also that E @jj, F is *-semisimple. Then, E, F are *-semisimple.
Proof. If (x,yjE E x F withx#O,y#O, thenx@y& R*(EhKF), i.e., there is d E R ‘(E hK F) with 0(x By) # 0. Thus, if dE, QF are the restrictions of $ to E, F, respectively, [6, Lemma 3.31, one has )(x @ y) = dE(x) br(y), so dE(x) # 0. d,(y) # 0, which yields the assertion by Proposition 4.4. m
5. COMMUTATIVE
*-SEMISIMPLE
We relate below, in the case of commutative semisimplicity considered before.
ALGEBRAS
algebras, the two kinds of
LEMMA 5.1. Let E be a commutative Imc *-algebra with a bai. Then. .9(E) = 9(E) n YIN(E), within a bijection. In other words, .9(E) = 2’ *(E). with Q:*(E) the Hermitian spectrum of E.
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MARIA FRAGOULOPOULOU
Proof: If E, is the respective unital algebra of E [5] and j?, its completion, one easily gets !Ui’(E) = W(E,) = %I@,) within bijections. Thus, the assertion follows by [l, Theorem 4.51 and [5, Proposition 3.41. 1 Remark 5.1. [4, Corollary 3.81 gives a Bochner-Weil-Raikov theorem for lmc *-algebras: that is, if E is a commutative lmc *-algebra with continuous Gel’fand map, in such a way that (x*)- = (X)-, x E E, then SEE’ is positive and extendable ifff(x) =,~(x^), x E E, with ,u a (complex) positive finite measure on !IR(E). (As a matter of fact we do not really need the continuity of the Gel’fand map). Now, if E has .a bai, everyfE 9(E) is extendable [5, Lemma 3.31, so that by restriction to g*(E), one gets the next abstract form of the above result:
Let E be a commutative lmc *-algebra with a bai. Also 1etfE E’. Then, f is positive iff f(x) = p(J), x E E, with p a positive finite measure on g*(E). Concerning the Banach *-algebras analogue of the latter (cf. [20, Corollary 9.141). In this regard, we still remark that Lemma 5.1 could also be derived from the preceding abstract form of Bochner-Weil-Raikov theorem. THEOREM 5.1. Let E be a commutative lmc *-algebra with a bai. Then, *-semisimplicity implies semisimplicity on E. Furthermore, the two notions coincide if E is, in particular, a commutative unital complete lmc C*-algebra with continuous Gel’fand map.
ProoJ
According to 120, Corollary 6.41 and Lemma 5.1 R’(E) = 2?*(E) =9(E)
E W(E),
so that n (ker(f), fE W(E)} c R*(E), which proves the first part of the assertion. Concerning the second part of the statement, the hypothesis implies that E = C,(!JJI(E))(the algebra of continuous complex-valued functions on W(E) with the topology of compact convergence) within a topological algebraic isomorphism (Gel’fand-Naimark theorem; cf. [ 19, Theorem 8.4; 18, Lemma IV, 3.21). Thus, setting X = 1132(E), X= %Jl(C,(X)) = G*(C,(X))
= R’(C,(X)),
within bijections the first being actually a homeomorphism 9.21. 1
[ 18, Theorem III,
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ACKNOWLEDGMENTS The author expresses her appreciation to Professor Anastasios Mallios for helpful and stimulating discussions during the preparation of this paper.
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