Let Ea(u,v) be the extremal algebra determined by two hermitians u and v with u2 = v2 = 1.  We show that: Ea(u,v) = {f=gu:f,g &#949; C(T)}, where T is the unit circle; Ea(u,v) is C*-equivelant to C*(G), where G is the infinite dihedral group; most of the hermitian elements k od Ea(u,v) have the property that kn is hermitian for all odd n but for no even n; any two hermitian words in G generate an isometric copy of Ea(u,v) in Ea(u,v)

Crabb, M.J.

Duncan, J.

McGregor, C.M.

English

M. J. Crabb

J. Duncan

C. M. McGregor

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The Extremal Algebra on Two Hermitians With Square 1

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The extremal algebra on two hermitians with square 1

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       Crabb, M.J. and Duncan, J. and McGregor, C.M. (2002) The extremal algebra on two hermitians with square 1. Glasgow Mathematical Journal, 44 (2). pp. 255-260. ISSN 0017-0895  http://eprints.gla.ac.uk/12918/  Deposited on: 19 April 2010   Enlighten – Research publications by members of the University of Glasgow http://eprints.gla.ac.uk THE EXTREMAL ALGEBRA ON TWO HERMITIANSWITH SQUARE 1M. J. CRABBDepartment of Mathematics, University of Glasgow, University Gardens, Glasgow G12 8QW, Scotlande-mail: m.crabb@maths.gla.ac.ukJ. DUNCAN*Department of Mathematical Sciences, SCEN301, University of Arkansas,Fayetteville, AR 72701-1201, USAe-mail: jduncan@comp.uark.eduand C. M. McGREGORDepartment of Mathematics, University of Glasgow, University Gardens, Glasgow G12 8QW, Scotlande-mail: c.mcgregor@maths.gla.ac.uk(Received 1 August, 2000; accepted 23 January 2001)Abstract. Let Eaðu; vÞ be the extremal algebra determined by two hermitians uand v with u2 ¼ v2 ¼ 1. We show that: Eaðu; vÞ ¼ f fþ gu : f; g 2 CðTÞg, where T isthe unit circle; Eaðu; vÞ is C-equivalent to CðGÞ, where G is the inﬁnite dihedralgroup; most of the hermitian elements k of Eaðu; vÞ have the property that kn ishermitian for all odd n but for no even n; any two hermitian words in G generate anisometric copy of Eaðu; vÞ in Eaðu; vÞ.2000 Mathematics Subject Classiﬁcation. 47A12, 47B15.1. Introduction. This is a continuation of [2], except that we are concerned hereonly with the extremal Banach algebra Eaðu; vÞ determined by two hermitian invo-lutions u and v (we use involution here in the group sense, namely that u2 ¼ v2 ¼ 1).In [2] we presented Eaðu; vÞ as an abstract completion of a group algebra. Here wepresent it as a speciﬁc algebra of pairs of continuous functions on the unit circle andwe prove that it is even C-equivalent for the natural star operation on Eaðu; vÞwhich makes the generators u and v unitary elements. The hermitian element deﬁnedby h ¼ ði=2Þðuv vuÞ has the remarkable property that hn is hermitian for every oddn but for no even n; and yet the subalgebra generated by h is C-equivalent toC½1; 1. The algebra Eaðu; vÞ is equivalent to the C-algebra of the inﬁnite dihedralgroup G. We give a simple explicit description of the space of hermitian elements inEaðu; vÞ; we also show that most of the hermitian elements k of Eaðu; vÞ have theproperty that kn is hermitian for all odd n but for no even n. Permutations of Ginduce (isometric) automorphisms of CðGÞ. We show that there are also many(isometric) isomorphisms onto subalgebras of Eaðu; vÞ.We use without comment some elementary properties of hermitians which maybe found in [1].Glasgow Math. J. 44 (2002) 255–260. # 2002 Glasgow Mathematical Journal Trust.DOI: 10.1017/S0017089502020062. Printed in the United Kingdom*This author acknowledges the support of a Scheme 2 Grant from the London Mathematical Society.2. Ea(u,v) is C*-equivalent. We repeat here some essential notation from [2]. Wewrite G for the inﬁnite dihedral group generated by x and u, where u2 ¼ 1 andux ¼ x1u. In relation to the algebra Eaðu; vÞ we have x ¼ uv. PutH ¼ fxn : n 2 Zg sothat G ¼ H [Hu. Put A0 ¼ C½G and recall the (algebra) involutions * and y given byXgg ¼Xgg1;Xgg y¼Xgg1:For a 2 A0 we note that a ¼ ay()a 2 R½G. Let J  C½G be the set of all ﬁniteproducts of elements of the form p ¼ cos  þ i sin xnu, where  2 R; n 2 Z. SinceðxnuÞ1 ¼ xnu, we have p ¼ cos   i sin xnu 2 J , and pp ¼ 1 ¼ pp. It followsthat, for all a 2 J , aa ¼ 1 ¼ aa and a 2 J . Hence J is a group in A0. Identitiessuch as cos  þ i sin  xu ¼ ðiuÞðcos  þ i sin  vÞðiuÞ show that this is the J of [2].Since xn ¼ ðixnuÞðiuÞ, we have xn 2 J and so H  J .Since G ¼ H [Hu, each a 2 A0 can be written as a ¼ bþ cu with b; c 2 C½H.For c 2 C½H, we have uc ¼ cyu. For b; c 2 C½H, this gives ðbþ cuÞ ¼ b þ cyu;deﬁne an involution z onA0 by ðbþ cuÞz ¼ by  cu. By contrast, note that ðbþ cuÞy ¼by þ cu. Then az ¼ az for a 2 A0, and a ¼ az if a 2 J . For a 2 A0, a ¼ az if andonly if a ¼ bþ icu for some b; c 2 R½H; then aa ¼ bbþ cc ¼ aa.Lemma 2.1. Let K ¼ fa 2 A0 : a ¼ az; aa ¼ 1g. Then J ¼ K.Proof. From the above, J  K. Clearly K is also a group. Any a 2 K,a 62 ðHUiHuÞ, may be writtena ¼ pxp þ    þ mxm þ iqxquþ    þ inxnu ð1Þwhere p; q;m; n 2 Z, p  m, q  n, pmqn 6¼ 0 and k; k 2 R for all k. Supposethat m p > n q. Then the coeﬃcient of xmp in aa is pm 6¼ 0. Since aa ¼ 1,this coeﬃcient is 0. Similarly we rule out n q > m p. Therefore m p ¼ n q;call this common value the length of a. We show that a 2 K implies a 2 J byinduction on the length of a. If a has length 0 then a ¼ pxp þ iqxqu ¼xpðp þ iqxqpuÞ 2 J , since 1 ¼ aa ¼ 2p þ 2q. Suppose that our claim holds forelements of length less than N, and consider a as above of length N. For  2 R, cos þi sin  xqpu 2 J  K and so a0 ¼ aðcos  þ i sin  xqpuÞ 2 K. Here a0 has the form ofð1Þwith p replaced by 0p ¼ p cos   q sin . We choose  so that 0p ¼ 0. Then a0 haslength less thanN and, by hypothesis, a0 2 J . Therefore a 2 J , as required. &As in [2], we now deﬁne a norm on A0 bykak ¼ infXN1jkj : a ¼XN1kak;N 2 N; k 2 C; ak 2 Jn o:Let T denote the unit circle in C. With each element b ¼Pnxn of C½H weassociate the function on T given by bðÞ ¼Pnn. We can now regard A0 as the setof all elements fþ gu where f; g are polynomials in  and 1 ¼  on T. We also havea representation as 2 2 matrices of functions of  2 T by	ð fþ guÞ ¼ fðÞ gðÞgðÞ fðÞ :256 M. J. CRABB, J. DUNCAN AND C. M. McGREGORThe involutions on C½H correspond tof ðÞ ¼ fðÞ; f yðÞ ¼ fðÞ:We have J ¼ J ¼ Jz, and so  and z are isometric for k  k. We write j  j1 for thesupremum norm over T. Of course the element x corresponds to the functionxðÞ ¼ .Lemma 2.2. Let f 2 R½H with j f j1 < 1. Then there exists g 2 R½H such thatffþ gg ¼ 1.Proof. Put F ¼ 1 f f, so that F is a positive trigonometric polynomial withreal coeﬃcients. By [3, pp 117–8], F can be written as gg, and the proof in [3] showsthat the trigonometric polynomial g also has real coeﬃcients. &Corollary 2.3. For f 2 R½H, we have k f k ¼ j f j1. For f 2 C½H, we havej f j1  k f k  2j f j1. The completion of ðC½H; k  kÞ is CðTÞ, with j f j1  k f k 2j f j1 for all f 2 CðTÞ.Proof. Let f 2 R½H with j f j1 < 1. By Lemma 2.2, there exists g 2 R½H suchthat f *fþ g*g ¼ 1. Then a ¼ f igu satisfy a ¼ az and aa ¼ 1. By Lemma 2.1,f igu 2 J . Therefore k f iguk ¼ 1, and k f k  1. By linearity, k f k  j f j1 forf 2 R½H. For bþ icu 2 J , we havejbðÞj2 þ jcðÞj2 ¼ ðbbþ ccÞðÞ ¼ 1and so jbðÞj  1 for  2 T. Hence, for f 2 C½H, k f k  j fðÞj, and so k f k  j f j1,which gives k f k ¼ j f j1 for f 2 R½H.Let f ¼Pnxn 2 C½H. Note that f *y ¼Pnxn, and j f yj1 ¼ j f j1. Thusfþ f *y 2 R½H and j fþ f yj1  2j f j1. This gives k fþ f *yk  2j f j1.Also, ið f f *yÞ 2 R½H, which gives k f f *yk  2j f j1 and hence k f k  2j f j1. Theﬁnal part follows by the Stone-Weierstrass theorem. &The involutions  and y extend in the natural way to CðTÞ, and a routineapproximation argument gives the next corollary. Deﬁne CSðTÞ ¼ ff 2 CðTÞ :f * ¼ f yg.Corollary 2.4. Let f 2 CSðTÞ. Then k f k ¼ j f j1.We deﬁne a norm j  j on C½G by jaj ¼ supfjabj2 : b 2 ‘2ðGÞ; jbj2 ¼ 1g, wherejPggj2 ¼ ðPjgj2Þ1=2. The completion of ðC½G; j  jÞ is the C-algebra CðGÞ.Lemma 2.5. Let L be a subgroup of G. Let a ¼Pg2G gg 2 C½G, andd ¼Pg2L gg its projection in C½L. Then jdj  jaj.Proof. Write a ¼ dþ f where f 2 linðG n LÞ. If b 2 ‘2ðLÞ then db 2 ‘2ðLÞ andfb 2 ‘2ðG n LÞ. Therefore jabj2 ¼ jdbþ fbj2  jdbj2. Taking the supremum overjbj2 ¼ 1, we have jaj  jdj. &THE EXTREMAL ALGEBRA ON TWO HERMITIANS WITH SQUARE 1 257Note that, with the notation of Lemma2.5, jdj is the samewhether taken overL orG.Theorem 2.6. As algebras,Eaðu; vÞ ¼ C*ðGÞ ¼ f fþ gu : f; g 2 CðTÞg;with uf ¼ f yu and j f j ¼ j f j1 ð f 2 CSðTÞÞ. For a 2 Eaðu; vÞ, jaj  kak  4jaj.Proof. Let a ¼ fþ gu with f; g 2 C½H. Lemma 2.5 gives j f j  jaj. Sinceau ¼ gþ fu, also jgj  jauj ¼ jaj. We have j f j ¼ j f j1. From the Stone-Weierstrasstheorem we deduce that C*ðGÞ ¼ f fþ gu : f; g 2 CðTÞg.It is now enough to prove that jaj  kak  4jaj for a ¼ fþ gu, f; g 2 C½H. Wehave that jaj  kak by the extremal nature of k  k. Also, kak  k f k þ kgk 2j f j þ 2jgj  4jaj by Corollary 2.3. &Corollary 2.7. The extremal Banach algebra on one generator with all oddpowers hermitian is C*-equivalent with j  j  k  k  2j  j where k  k is the extremalnorm and j  j the C*-norm.We extend  and z to Eaðu; vÞ by the earlier formulæ. For the above matrixrepresentation, z gives the adjugate matrix.3. Properties of Ea(u,v). We begin by identifying the space of hermitian ele-ments in Eaðu; vÞ. In [2] we noted the obvious hermitian elements (in A0) given byxnu (n 2 Z), 1 and iðxn  xnÞ (n 2 N). As expected, the space H of hermitian ele-ments of Eaðu; vÞ is the closed real linear span of these elements. In fact, we can givea more elegant, and useful, description in terms of the involutions  and z.Theorem 3.1. We have H ¼ fh 2 Eaðu; vÞ : h* ¼ h; hþ hz 2 Rg.Proof. Suppose that h 2 Eaðu; vÞ with h* ¼ h and hþ hz ¼  2 R. Replacing hby h =2, we assume that  ¼ 0. We approximate h by elements k in A0 satisfyingk ¼ k* ¼ kz. We verify that k is a real linear combination of elements xnu andiðxn  xnÞ for n 2 Z. Hence k, and so its limit h, is hermitian.Now suppose that h 2 H. By extremality, h is also hermitian in CðGÞ, and soh ¼ h. Let  2 T and  2 C. Deﬁne a linear functional  on A byðbþ cuÞ ¼ ð1 2Þbð1Þ þ bðÞ þ bðÞ ðb; c 2 CðTÞÞ:Then ð1Þ ¼ 1. If bþ cu 2 J then, as in Corollary 2.3, 1  bð1Þ  1, jbðÞj  1 andbðÞ ¼ bðÞ. These give jðbþ cuÞj  maxf1; j1 4jg. If j1 4j  1 then jðJ Þj  1and so kk  1. For these ,  is a support functional of 1. Hence ðhÞ 2 R. Writeh ¼ fþ guwith f; g 2 CðTÞ. We deduce that fð1Þ 2 R and fðÞ þ fðÞ ¼ 2fð1Þ. Thereforehþ hz ¼ fþ f z ¼ 2fð1Þ, as required. &The proof of the next result is routine.258 M. J. CRABB, J. DUNCAN AND C. M. McGREGORProposition 3.2. The centre Z of Eaðu; vÞ is given by Z ¼ ff 2 CðTÞ : f ¼ f yg andZ \H ¼ R.We show that most hermitian elements h of Eaðu; vÞ have the property that hn ishermitian for all odd n but for no even n. On the other hand, when h contains a non-zero multiple of the identity, we usually have no other power hermitian. We remarkthat these latter hermitians cannot generate the extremal algebra on one hermitiangenerator because they generate C-equivalent subalgebras.Let H0 ¼ fh 2 A : h ¼ h ¼ hzg, so that H0 # H.Theorem 3.3 Let n 2 N.(1) If h 2 H0 and n is odd then hn 2 H.(2) If either h 2 H0 and hn 2 H with n even, or h 2 H nH0 and hn 2 H withn > 1, then PðhÞ ¼ 0 for some quadratic polynomial P.Proof. (1) Since h ¼ h ¼ hz, we have hn ¼ hn ¼ hnz for n odd, and sohn 2 H0.(2) For some ;  2 R, hþ hz ¼  and hn þ hzn ¼ , where  ¼ 0 and n is even, or 6¼ 0 and n > 1. Consider the even polynomialQðÞ ¼ n þ ð Þn  , which has atmost two real zeros. Then QðhÞ ¼ 0, and each factor h  of QðhÞ with  62 R may becancelled since h has real spectrum. This leaves a real quadratic P with PðhÞ ¼ 0. &An example of the situation in Theorem 3.3 (2) is h ¼ iðx x1Þ þ ðxþ x1Þu.Here h 2 H0 and h2 ¼ 4. In these cases, hn 2 H ðn 2 NÞ.The inﬁnite dihedral group G has many subgroups which are isomorphic to Gand hence the C-algebra generated by such is isometrically isomorphic to CðGÞ.There are natural related questions to ask for Eaðu; vÞ. Since kuk ¼ ku1k ¼ 1, themapping a! uau is an isometric monomorphism of Eaðu; vÞ. Thus the closed sub-algebra generated by u; uvu is a copy of Eaðu; vÞ. Equally for the closed subalgebragenerated by vuv; v. By applying these two mappings repeatedly we easily see thatthe closed subalgebra generated by xnu; xnþ1u is a copy of Eaðu; vÞ for any n 2 Z. Onthe other hand, this simple method will not identify for us the closed subalgebragenerated by uvu; vuv (i.e. xu; x2u). We show in fact that any two hermitian ele-ments xmu; xnu with m; n 2 Z;m 6¼ n generate a copy of Eaðu; vÞ.Let AS ¼ fa 2 Eaðu; vÞ : a ¼ azg. We easily verify that AS ¼ f fþ igu :f; g 2 CSðTÞg. Also, AS is a real C-algebra with the involution  and norm j  j.Proposition 3.4. We have kak ¼ jaj for a 2 AS.Proof. Let a 2 AS with jaj < 1. By [4], a is a convex combination of elements ofthe form cos b ec, where b; c 2 AS, b* ¼ b, c* ¼ c. Then b 2 CSðTÞ, b is real valued,cos b 2 CSðTÞ and so k cos bk ¼ j cos bj1  1. Also, ðicÞ* ¼ ic* ¼ ic ¼ ðicÞz andso ic 2 H0, keck ¼ 1. Therefore kak  1. It follows that kak  jaj for all a 2 AS. Butjaj  kak by Theorem 2.6. Hence kak ¼ jaj. &Theorem 3.5. Let xmu, xnu be any two hermitian words in G (where m; n 2 Z).Then they generate an isometric copy of Eaðu; vÞ in Eaðu; vÞ.THE EXTREMAL ALGEBRA ON TWO HERMITIANS WITH SQUARE 1 259Proof. In G, xmu and xnu generate an isomorphic subgroup G1 ¼ H1 [ K1, whereH1  H and K1  Hu. Deﬁne a norm k  k1 on C½G1 via J 1 ¼ fa 2 C ½G1 :a* ¼ az; a*a ¼ 1g. The involutions , z of C½G1 agree with those of C½G. ThenðC½G1; k  k1Þ is an isometric copy of A0. It is enough to show that k  k1 is just therestriction of the norm k  k of A0.Let a 2 J , and d its projection in C½G1. By Lemma 2.5, jdj  jaj ¼ 1. Froma* ¼ az we deduce that d* ¼ dz. By Proposition 3.4, kdk1 ¼ jdj. Hence kdk1  1.Now consider a 2 C½G1. Suppose a ¼Pkak with k 2 C and ak 2 J . Let dk bethe projection of ak inC½G1. Then kak1 ¼ kPkakk1 ¼ kPkdkk1 Pjkj. It followsthat kak1  kak. But since J 1  J , kak1  kak. Therefore kak1 ¼ kak. &REFERENCES1. F. F. Bonsall and J. Duncan, Numerical ranges of operators on normed spaces andelements of normed algebras, London Math. Soc. Lecture Note Series No 2, (CambridgeUniversity Press, 1971).2. M. J. Crabb, J. Duncan and C. M. McGregor, Some extremal algebras forhermitians, Glasgow Math. J. 43 (2001), 29–38.3. F. Riesz and B. Sz.-Nagy, Functional analysis, (Frederick Unger Pub. Co., NewYork, 1955).4. B. R. Li, Real operator algebras, Scientia Sinica 22 (1979), 733–746.260 M. J. CRABB, J. DUNCAN AND C. M. McGREGOR

The extremal algebra on two hermitians with square 1

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