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1010.Full.Pdf TRANSLATION OPERATORS ON THE HALF-LINE BY L. A. COBURN* AND R. G. DOUGLAS* THE BELFER GRADUATE SCHOOL, YESHIVA UNIVERSITY, AND THE UNIVERSITY OF MICHIGAN Communicated by Alberto P. Calder6n, December 19, 1968 Abstract.-The self-adjoint algebra of operators generated by the semigroup of translation operators acting on the Hilbert space of functions supported on the half-line is studied. A real-valued index is introduced and is used to determine the spectrum of the Wiener-Hopf integral operators with distribution kernel having an almost periodic Fourier transform. Further, the algebra is shown to contain no nonzero compact operators, and the quotient of the algebra by its commutator ideal is shown to be isometrically isomorphic to the Banach algebra of almost periodic functions on the line. Let R denote the additive group of real numbers and L' (R) the usual Hilbert space of complex-valued, Lebesgue square-integrable functions on R. Further, let L2(R+) denote the (closed) subspace of L2(R) consisting of the functions supported on the semigroup R+ of nonnegative real numbers. We consider the one-parameter semigroup = Tx: X:C R+} of isometries on L2(R+) defined by (Tf)(x) = f(x - X). In this note we describe an analysis of the C*-algebra P generated by S. The algebra P is in many ways similar to certain algebras of singular integral operators,1 and an index theory is crucial in our results. The index group in this case, however, is the discrete group Rd of real numbers. The structure of P proves to be analogous to that of the C*-algebra generated by a single isometry,2 except that in this instance P contains no nonzero compact operator. The algebra P does contain the multiplication operators by CO functions with compact support and thus contains the difference operators on R+ having smooth coefficients with limits at infinity. A further interest in P stems from the fact that it contains the Wiener-Hopf operators with distribution kernels having an almost periodic Fourier trans- form. Moreover, our analysis of P allows us to determine the spectrum of such an operator. Lastly, since this class of operators can also be defined as singular integral operators,4 we have an instance of an "index theorem" in which the topological invariant is a real number rather than an integer. Our analysis makes use of the usual Fourier transform F on L2(R). We begin by recalling that F-1 maps L2(R+) onto the subspace H2 of L2(R) con- sisting of boundary values of functions analytic in the upper half-plane. It is clear that for X in R+, the operators (Mexp(iXxj)f)(x) = eiX f(x) are isometries on H2, and 1010 Downloaded by guest on September 24, 2021 VOL. 62, 1969 MATHEMATICS: COBURN AND DOUGLAS 1011 F-'T),F = Mep(i\x). Thus, we have a useful, spatially equivalent way of representing R. LEMMA 1. The closed two-sided ideal C generated by the commutators in P is the closed span of all the finite sums N 2 MviTaiy i=1 where (pi is a function of bounded variation with compact support in R+, and ai is in R with = f(x-a) x > 0 (Taf)(x) (0 X < 0. Sketch of proof: It is easy to see that T* = T-a. It is also easy to check that C contains MV,, where so is the characteristic function of any open finite interval in R+. The result follows from standard approximation techniques. Now let P denote the orthogonal projection from L'(R) onto H2. For so any (continuous) almost periodic function on R, we consider the operator WP = FPMF-' acting on L2(R+). Let 7r be the quotient map from P onto /IC. LEMMA 2. If p is an almost periodic function on R, then IIW,PII = infllW,, + CG1 = Ik(plK. THEOREM 1. The algebra A contains W, for each almost periodic function (p on R and every element in i can be written uniquely in the form W,, + C, where (p is an almost periodic function on R and C is in the commutator ideal 6. Sketch of proof: That W, is contained in A for every almost periodic function (p on R follows from the observation that for all real X Tx = Wexp(ixx), and the fundamental approximation theorem for almost periodic functions. The unique decomposition follows from an analysis of the generators of A and Lemma 2. We note that Theorem 1 suggests that the function s plays the role of a symbol' in the algebras of singular integral operators. This is made more precise by THEOREM 2. The mapping 7r(W,,) <-> so gives rise to a *-isomorphim between PIC and the algebra A of almost periodic functions on R with supremum norm. We can now consider the (abstract) Fredholm elements' of A with respect to the ideal C, Fred(RC) = {T C R:ir(T) invertible in P/C}C. The ideal C consists roughly of those operators in A which "vanish at infinity." Clearly, Fred(A,C) is an open subset of A that is closed under multiplication and is invariant under perturbations from C. Although the elements of FredQ(,C) do not admit a simple characterization in terms of null-space, closed range, etc., the components of Fred(R,C). can be described. Downloaded by guest on September 24, 2021 1012 MATHEMATICS: COBURN AND DOUGLAS PROc. N. A. S. THEOREM 3. The components of Fred(R,e) admit a group structure compatible with operator multiplication. This group is naturally isomorphic to Rd. Sketch of proof: The components of Fred(P,C) are in a natural one-to-one correspondence with the components of the group G of invertible elements in P/C. The latter collection of components can be identified with the group GIG,, where G1 is the component of the identity in G. Using Theorem 2 and a result of Royden,7 we see that G/G, is isomorphic to the first Cech cohomology group with integer coefficients of the maximal ideal space iMA of A. It is well known that iMA is the Bohr compactification RB of the reals. Using a classical result of Bohr8 or computing directly, one can verify that H1(RB,Z) is naturally isomorphic to Rd.9 We can now define a real-valued index K on Fred(R,6) as follows: for X in Fred(R,C), we let K(X) be the real number associated with the cohomology class that corresponds to the component of Fred(R,e) containing X. It follows immediately that K has the following properties: (1) K(XY) = K(X) + K(Y), (2) K(X+C) = K(X) for C in e, and (3) K iS continuous. One can further verify that To is in Fred(R,C) and, by using the naturalness of the isomorphisms involved, that K(Tx) = X. Thus K is a natural index on Fred(R,(?). As a consequence of Theorem 3, we have a COROLLARY. For sp an almost periodic function on R, the operator W,, is invertible if and only if W, is in Fred(R,C) and K(W,) = 0. The preceding can be used to determine the invertibility of the operator Wi,, since W8, is in Fred(R,,) if and only if (p is bounded away from 0 and since a real-valued index Kt called the "mean motion" may be defined'0 directly on the almost periodic functions so that Kt((p) = K(W,,). Now let K be the algebra of all compact operations on L2(R+). In contrast to the usual algebras of singular integrals, we have THEOREM 4. R n K = (0). Sketch of proof: Since R is an irreducible C*-algebra, it follows from known results that either K C C or i n C = (0). To show that K C C is impossible, we use Lemma 1 together with a suggestion by C. Berger and a result of Berger and Coburn which states that the C*-algebra generated by the translation operators and LO multipliers on LI of the circle group contains no nonzero compact operators." As an immediate consequence of Theorem 4 and a known result,'2 we see that if an operator X in R is Fredholm in the usual sense, that is, X has finite- dimensional null and defect spaces and closed range, then X must be invertible. In particular, this observation applies to difference operators on R+ with smooth coefficients having limits at infinity. Portions of our results admit generalization. Most of our results hold for other one-parameter semigroups of nonunitary isometric operators. Further, the group R can be replaced by a locally compact abelian group and R+ by a "sufficiently large" sub-semigroup to obtain most of the same results. Downloaded by guest on September 24, 2021 VOL. 62, 1969 MATHEMATICS: COBURN AND DOUGLAS 1013 * Research by L. A. C. supported by NSF grant GP-7520. Research by R. G. D. sup- ported by the National Science Foundation and Sloan Foundation. 1Calderon, A. P., and A. Zygmund, "Singular integral operators and differential equations," Am. J. Math., 79, 901-921 (1957). 2 Coburn, L. A., "The C*-algebra generated by an isometry I, II," Bull. Am. Math. Soc., 73, 722-726 (1967); Trans Am. Math. Soc., 137, 122-133 (1969). ' Cf. Douglas, R. G., "Toeplitz and Wiener-Hopf operators in H- + C," Bull. Am. Math. Soc., 74, 895-899 (1968). 4 Cf. Pincus, J. A., "The spectral theory of self-adjoint Wiener-Hopf operators," Bull. Am. Math. Soc., 72, 882-887 (1966).
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