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Universality of Some C ∗-Algebra Generated by a Unitary and a Self

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Universality of Some C ∗-Algebra Generated by a Unitary and a Self Universality of some C∗-algebra generated by a unitary and a self-adjoint idempotent Dedicated to Nikolai Vasilevski on the occasioin of his 60th birthday Helena Mascarenhas and Bernd Silbermann1 Keywords: C∗-algebra, nitely generated, universality Abstract We prove that there is essentially only one C∗-algebra generated by a unitary element u and a self-adjoint idempotent p such that up = pup and up 6= pu . This result is related to a theorem of L. Coburn stating that there is essentially only one C∗-algebra generated by a non-unitary isometry. 1 Introduction A theorem of L. Coburn [?] tells that the C∗-algebra generated by a non- unitary isometry v (that is v∗v = e, vv∗ 6= e) is universal in the sense that any two C∗-algebras with this property are isometrically isomorphic. An instructive example for such a C∗-algebra is given as follows: Let H2 ⊂ L2(T), T := {z ∈ C : |z| = 1}, be the familiar Hardy space and P + : L2(T) → H2 the Riesz projection which is known to be selfadjoint and surjective. Given a ∈ L∞(T), dene the operator T (a): H2 → H2 , f 7→ P +af . The operator T (a) is clearly bounded, and is called a Toeplitz operator. Let be the functions ±1 . Then ∗ and χ±1 χ±1(t) = t , t ∈ T T (χ1) = T (χ−1) ∗ ∗ T (χ1)T (χ1) = I, but T (χ1)T (χ1) 6= I. Hence, T (χ1) is a non-unitary isometry. It is well-known that the smallest C∗-algebra T (C) ⊂ B(H2) 1Mathematics Subject Classication (2000) 46L05 1 generated by T (χ1) (B(X) stands for the Banach algebra of all bounded linear operators acting on the Banach space X) contains all Toeplitz operators T (a), with a ∈ C(T), where C(T) denotes the algebra of all continuous functions on T. Moreover, 2 T (C) = {T (a): a ∈ C(T)}+˙ K(H ) , where K(H2) denotes the ideal of all compact operators acting on H2 (see for instance [?], Corollary 4.15 and Theorem 4.24). Clearly, is a unitary operator on 2 ∗ , and + is a χ1I L (T), (χ1I) = χ−1I P selfadjoint projection. It is easy to see that + + + + + χ1P = P χ1P and χ1P 6= P χ1I. The smallest ∗-subalgebra 2 containing and + con- C SO(C) ⊂ B(L (T)) χ1I P tains all singular integral operators A = aP + + bP − with P − := I − P + and a, b ∈ C(T). Moreover, + + − − 2 SO(C) = {P aP + P bP : a, b ∈ C(T)}+˙ K(L (T)) , where K(L2(T)) is again the ideal of all compact operators acting on L2(T) (see [?], Corollary 4.76; notice that this corollary is applicable in our case, and that QC(U) can be identied with K(L2(T))). These considerations show that T (C) is a subalgebra of SO(C). The close relationship between Toeplitz and singular integral operators gives rise to the question whether SO(C) is a model for a universal C∗-algebra. We will show that this is indeed the case. Our proof is based on features known from the theory of singular integral operators with continuous coecients. This theory is developed in many text books, and we will use mainly [?] and [?] because the approach given there is mostly convenient for us. Let us also notice that the study of nitely generated algebras is an important task. More about this topic can be found, in particular, in [?] and [?]. 2 2 The main result and its proof Let A be any C∗-algebra generated by a unitary element u (that is uu∗ = u∗u = e) and by a selfadjoint idempotent p+ such that up+ = p+up+ und up+ 6= p+u . (1) Passing to adjoints and using u−1 = u∗ yields p+u−1 = p+u−1p+ , (2) and thus u−1p− = p−u−1p− , where p− := e − p+. We denote by P the class of all C∗-algebras of this type. ∗ P Theorem: Let A1 and A2 be arbitrary C -algebras belonging to . Then these algebras are isometrically isomorphic. This isomorphism can be chosen so that u1, p1 are taken into u2, p2, respectively. Proof: Due to the GNS-construction we may assume (without loss of gen- erality) that there is a Hilbert space H such that for A ∈ P the elements u and p+ are operators on H which generate A. Let q ∈ C(T) be an arbitrarily given quasipolynomial, k k X j and form X j q(t) = ajt , aj ∈ C q(u) = aju . j=−k j=−k We call q(u) ∈ A a quasipolynomial of u and let L0(u) stand for the (non- closed) algebra of all quasipolynomials of u, and let L(u) be the closure of L0(u) in B(H). Since u is a unitary element, we know that −1 sp(u) = sp(u ) = T . The general theory of commutative C∗-algebras entails that L(u) is isomet- rically isomorphic to , and the isomorphism takes into : C(T) u χ1 ∼ L(u) = C(T) . 3 Now let A ∈ P be generated by u and p+. Introduce the (dense) subalgebra Z of A: ( M N ) X Y + − Z := (avs(u)p + bvs(u)p ): avs(u), bvs(u) ∈ L0(u), M, N ∈ N . v=1 s=1 Take an element c ∈ Z, say M N X Y + − c := (avs(u)p + bvs(u)p ) v=1 s=1 and suppose that it is invertible in A. We would like to express its invertibility in terms which can be used for further analysis. For this aim we proceed as follows: Given let r stand for . r ∈ N H {(h1, . , hr): hj ∈ H, j = 1, . , r} This linear space becomes a Hilbert space by introducing the scalar product r X h(h1, . , hr), (g1, . , gr)i = hhj, gji , j=1 and B(Hr) can be identied with (B(H))r×r in a natural way. Then form a linear extension c˜ of c exactly as it is done in [?], Chapter VIII, 10. The properties of c˜ are (and only these are needed): 1. c˜ ∈ (B(H))r×r for some r ∈ N and the entries of c˜ are among the + − + − elements O, I, a11(u)p + b11(u)p , . , aMN (u)p + bMN (u)p . 2. c is invertible in B(H) (and therefore in A by the inverse closedness property of unital C∗-subalgebras) if and only if c˜ is invertible in B(Hr). Now observe that c˜ can be written by 1. as + + − − c˜ = A1(u)diag{p , . , p } + A2(u)diag{p , . , p } , | {z } | {z } r−times r−times where A1(u),A2(u) are r×r-matrices with entries from L0(u). Now we apply the theory of [?], Chapter VIII, 8 (the condition dim coker (u|imP +) < ∞, which is formulated there, can be droped down because we are only interested in invertibility): 4 a) the invertibility of c˜ implies the invertibility of A1(u) and A2(u), that is for all detA1(t) 6= 0, detA2(t) 6= 0 t ∈ T , where A1(t) and A2(t) are the matrices whose entries are the Gelfand transforms of the entries of A1(u) and A2(u), respectively. r×r b) Notice that the functions A1(t),A2(t) belong to the class W , where W is the Wiener algebra. Moreover, the invertibility of c˜ implies that the right canonical Wiener-Hopf factorization of −1 t 7→ A2 (t)A1(t)(∈ W r×r) exists with all partial indices equal to zero. Conversely, if this condition is fullled then c˜, and thus also c, is invertible. Notice that this fact is true independently of the choice of A. Now take two algebras P A1 and A2 from . The element c and the algebra Z are denoted now by c1, c2 and Z1,Z2, respectively. Further, we dene a map ψ : Z1 → Z2, c1 7→ c2 and we need to show that ψ is correctly dened. Using that c1 is invertible if and only if c2 is invertible (in Ai) then by the above argument, we get sp(c1) = sp(c2) . Because ∗ ∗ , we therefore obtain c1c1 ∈ Z1, c2c2 ∈ Z2 ∗ ∗ sp(c1c1) = sp(c2c2) , and using 2 ∗ ∗ 2, that kc1k = kc1c1k = kc2c2k = kc2k kc1k = kc2k . (Recall that for a selfadjoint element the norm and the spectral radius coincide.) This equality shows that ψ is correctly dened and that ψ represents an isometric isomorphism between Z1 and Z2. Moreover, + +. The continuous extension of to the whole of ψu1 = u2, ψp1 = p2 ψ A1 provides us with the wanted isomorphism and nishes the proof. Remark: This proof can easily be adapted to prove Coburns result which was originally proved by dierent methods. Acknowledgement. First author partially supported by CEAF, IST, Tech- nical Univ. of Lisbon, and Fundação para a Ciência e a Tecnologia through the program FCT/FEDER/POCTI/MAT/59972/2004. 5 References [1] L. Coburn: The C∗-algebra generated by an isometry. Bull. Amer. Soc. 73 (1967), 722-726. [2] I. Gohberg, I. Feldman: Convolution Equations and Projection Methods for Their Solution. Nauka, Moskva 1971 (Russian) [Engl. translation: Transl. of Math. Monographs 41, Amer. Math. Soc., Providence, R.I., 1974]. [3] S. Pröÿdorf, B. Silbermann: Numerical Analysis for Integral and Related Operator Equations. Akademie-Verlag Berlin 1991, and Birkhäuser Verlag, Basel-Boston-Stuttgart, 1991. [4] V. Ostrovskyi, Y. Samoilenko: Introduction to the theory of representations of nitely presented ∗-algebras. I. Representations by bounded operators, Harwood Academic Publishers, Amsterdam, 1999. [5] N. Vasilevskii: C∗-algebras generated by orthogonal projections and their applications. IEOT 31 (1998), no. 1, 113-132.
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