×

A \(C^\ast\)-algebra of singular integral operators with shifts admitting distinct fixed points. (English) Zbl 1316.47063

The authors deal with a nonlocal \(C^*\)-algebra \(\mathfrak B\) generated by a \(C^* \)-algebra of singular integral operators with piecewise slowly oscillating coefficients and by a group of unitary shift operators \(U_{G}\) associated with a discrete amenable group \(G\) of orientation-preserving piecewise smooth homeomorphisms. Let \(\mathrm{PSO}(\mathbb T)\) be the \(C^*\)-algebra of piecewise slowly oscillating functions on the unit circle \(\mathbb T\) in \(\mathbb C\) and let \(G\) be a discrete amenable group of orientation-preserving piecewise smooth homeomorphisms \(g:\mathbb T\to\mathbb T\) possessing derivatives \(g^{\prime}\) with at most finite sets of discontinuities which act on \(\mathbb T\) topologically freely. The authors consider the \(C^*\)-subalgebra \(\mathfrak B=\mathrm{alg}(\mathrm{PSO}(\mathbb T), S_{\mathbb T},U_G)\) of \(\mathcal B(L^2(\mathbb T))\), the \(C^*\)-algebra of all bounded linear operators on \(L^2(\mathbb T)\), generated by all multiplication operators \(aI\) with \(a\in \mathrm{PSO}(\mathbb T)\), by the Cauchy singular integral operator \(S_{\mathbb T}\) defined by \[ (S_{\mathbb T}\varphi)(t)=\lim\limits_{\varepsilon \to 0} \frac{1}{\pi i}\int\limits_{\mathbb T\smallsetminus\mathbb T(t,\varepsilon)}\frac{\varphi(\tau)}{\tau-t}d\tau,\;\;\;\mathbb T(t,\varepsilon)=\{\tau\in\mathbb T: |\tau-t|<\varepsilon\},\quad t\in\mathbb T, \] and by the group \(U_G=\{u_g:g\in G\}\) of unitary weighted shift operators \(U_g\) given by \((U_g\varphi)(t)=|g^{\prime}(t)|^{\frac{1}{2}}\varphi(g(t))\) for all \(t\in\mathbb T\). They construct a representation \(\Psi_{\mathfrak B}\) of \(\mathfrak B\) on a suitable Hilbert space \(\mathcal H_{\mathfrak B}\). A Fredholm symbol map for \(\mathfrak B\) is obtained, that is, \(B\in\mathfrak B\) is Fredholm if and only if \(\Psi_{\mathfrak B}(B)\) is invertible on \(\mathcal H_{\mathfrak B}\). This gives a faithful representation of the quotient \(C^*\)-algebra \(\mathfrak B/\mathcal K\) on a suitable Hilbert space, where \(\mathcal K\) is the ideal of compact operators of \(\mathcal B(L^2(\mathbb T))\).
Reviewer: Guoxing Ji (Xian)

MSC:

47L80 Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)
46L05 General theory of \(C^*\)-algebras
47A67 Representation theory of linear operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Antonevich, A., Linear Functional Equations. Operator Approach, Oper. Theory Adv. Appl., vol. 83 (1988), Birkhäuser: Birkhäuser Basel: University Press: Birkhäuser: Birkhäuser Basel: University Press Minsk, Russian original: · Zbl 0708.47010
[2] Antonevich, A.; Lebedev, A., Functional Differential Equations: I. \(C^\ast \)-Theory, Pitman Monogr. Surv. Pure Appl. Math., vol. 70 (1994), Longman: Longman Harlow
[3] Bastos, M. A.; Fernandes, C. A.; Karlovich, Yu. I., \(C^\ast \)-algebras of integral operators with piecewise slowly oscillating coefficients and shifts acting freely, Integral Equations Operator Theory, 55, 19-67 (2006) · Zbl 1124.47010
[4] Bastos, M. A.; Fernandes, C. A.; Karlovich, Yu. I., Spectral measures in \(C^\ast \)-algebras of singular integral operators with shifts, J. Funct. Anal., 242, 86-126 (2007) · Zbl 1108.47012
[5] Bastos, M. A.; Fernandes, C. A.; Karlovich, Yu. I., \(C^\ast \)-algebras of singular integral operators with shifts having the same nonempty set of fixed points, Complex Anal. Oper. Theory, 2, 241-272 (2008) · Zbl 1158.47066
[6] Bastos, M. A.; Fernandes, C. A.; Karlovich, Yu. I., A \(C^\ast \)-algebra of functional operators with shifts having a nonempty set of periodic points, (Begehr, H. G.W.; etal., Further Progress in Analysis. Proceedings of the 6th International ISAAC Congress. Further Progress in Analysis. Proceedings of the 6th International ISAAC Congress, Ankara, Turkey, August 13-18, 2007 (2009), World Scientific: World Scientific Hackensack, NJ), 111-121 · Zbl 1207.47082
[7] Bastos, M. A.; Fernandes, C. A.; Karlovich, Yu. I., A nonlocal \(C^\ast \)-algebra of singular integral operators with shifts having periodic points, Integral Equations Operator Theory, 71, 509-534 (2011) · Zbl 1267.47015
[8] Böttcher, A.; Karlovich, Yu. I.; Spitkovsky, I. M., Convolution Operators and Factorization of Almost Periodic Matrix Functions, Oper. Theory Adv. Appl., vol. 131 (2002), Birkhäuser: Birkhäuser Basel · Zbl 1011.47001
[9] Böttcher, A.; Roch, S.; Silbermann, B.; Spitkovsky, I. M., A Gohberg-Krupnik-Sarason symbol calculus for algebras of Toeplitz, Hankel, Cauchy, and Carleman operators, (de Branges, L.; Gohberg, I.; Rovnyak, J., Topics in Operator Theory. Ernst D. Hellinger Memorial Volume. Topics in Operator Theory. Ernst D. Hellinger Memorial Volume, Oper. Theory Adv. Appl., vol. 48 (1990), Birkhäuser: Birkhäuser Basel), 189-234 · Zbl 0727.47037
[10] Böttcher, A.; Silbermann, B., Analysis of Toeplitz Operators (2006), Springer: Springer Berlin
[11] Douglas, R. G., Banach Algebra Techniques in Operator Theory (1972), Academic Press: Academic Press New York · Zbl 0247.47001
[12] Gohberg, I.; Krupnik, N., On the algebra generated by the one-dimensional singular integral operators with piecewise continuous coefficients, Funct. Anal. Appl., 4, 193-201 (1970) · Zbl 0225.45005
[13] Greenleaf, F. P., Invariant Means on Topological Groups and Their Representations (1969), Van Nostrand-Reinhold: Van Nostrand-Reinhold New York · Zbl 0174.19001
[14] Hagen, R.; Roch, S.; Silbermann, B., Spectral Theory of Approximation Methods for Convolution Equations (1995), Birkhäuser: Birkhäuser Basel
[15] Karlovich, Yu. I., The local-trajectory method of studying invertibility in \(C^\ast \)-algebras of operators with discrete groups of shifts, Soviet Math. Dokl., 37, 407-411 (1988) · Zbl 0695.47029
[16] Karlovich, Yu. I., \(C^\ast \)-algebras of operators of convolution type with discrete groups of shifts and oscillating coefficients, Soviet Math. Dokl., 38, 301-307 (1989) · Zbl 0705.47043
[17] Karlovich, Yu. I., On algebras of singular integral operators with discrete groups of shifts in \(L_p\)-spaces, Soviet Math. Dokl., 39, 48-53 (1989) · Zbl 0732.47044
[18] Karlovich, Yu. I., A local-trajectory method and isomorphism theorems for nonlocal \(C^\ast \)-algebras, (Erusalimsky, Ja. M.; Gohberg, I.; Grudsky, S. M.; Rabinovich, V.; Vasilevski, N., Modern Operator Theory and Applications. The Igor Borisovich Simonenko Anniversary Volume. Modern Operator Theory and Applications. The Igor Borisovich Simonenko Anniversary Volume, Oper. Theory Adv. Appl., vol. 170 (2007), Birkhäuser: Birkhäuser Basel), 137-166 · Zbl 1120.46052
[19] Karlovich, A. Yu.; Karlovich, Yu. I.; Lebre, A. B., Invertibility of functional operators with slowly oscillating non-Carleman shifts, (Böttcher, A.; Kaashoek, M. A.; Lebre, A. B.; dos Santos, A. F.; Speck, F.-O., Singular Integral Operators, Factorization and Applications. International Workshop on Operator Theory and Applications. Singular Integral Operators, Factorization and Applications. International Workshop on Operator Theory and Applications, IWOTA 2000, Portugal. Singular Integral Operators, Factorization and Applications. International Workshop on Operator Theory and Applications. Singular Integral Operators, Factorization and Applications. International Workshop on Operator Theory and Applications, IWOTA 2000, Portugal, Oper. Theory Adv. Appl., vol. 142 (2003), Birkhäuser: Birkhäuser Basel), 147-174 · Zbl 1027.47025
[20] Karlovich, Yu. I.; Kravchenko, V. G., An algebra of singular integral operators with piecewise-continuous coefficients and piecewise-smooth shift on a composite contour, Math. USSR-Izv., 23, 307-352 (1984) · Zbl 0565.47009
[21] Karlovich, Yu. I.; Silbermann, B., Local method for nonlocal operators on Banach spaces, (Böttcher, A.; Gohberg, I.; Junghanns, P., Toeplitz Matrices and Singular Integral Equations. The Bernd Silbermann Anniversary Volume. Toeplitz Matrices and Singular Integral Equations. The Bernd Silbermann Anniversary Volume, Oper. Theory Adv. Appl., vol. 135 (2002), Birkhäuser: Birkhäuser Basel), 235-247 · Zbl 1041.47005
[22] Karlovich, Yu. I.; Silbermann, B., Fredholmness of singular integral operators with discrete subexponential groups of shifts on Lebesgue spaces, Math. Nachr., 272, 55-94 (2004) · Zbl 1073.47049
[23] Naimark, M. A., Normed Algebras (1972), Wolters-Noordhoff: Wolters-Noordhoff Groningen
[24] Power, S. C., Fredholm Toeplitz operators and slow oscillation, Canad. J. Math., 32, 1058-1071 (1980) · Zbl 0456.47024
[25] Roch, S.; Santos, P. A.; Silbermann, B., Non-Commutative Gelfand Theories. A Tool-kit for Operator Theorists and Numerical Analysts (2011), Springer: Springer London · Zbl 1209.47002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.