Matrix bundles and operator algebras over a finitely bordered Riemann surface. (English) Zbl 1480.47113
Summary: This note presents an analysis of a class of operator algebras constructed as cross-sectional algebras of flat holomorphic matrix bundles over a finitely bordered Riemann surface. These algebras are partly inspired by the bundle shifts of M. B. Abrahamse and R. G. Douglas [Adv. Math. 19, 106–148 (1976; Zbl 0321.47019)]. The first objective is to understand the boundary representations of the containing \(C^*\)-algebra, i.e., Arveson’s noncommutative Choquet boundary for each of our operator algebras. The boundary representations of our operator algebras for their containing \(C^*\)-algebras are calculated, and it is shown that they correspond to evaluations on the boundary of the Riemann surface. Secondly, we show that our algebras are Azumaya algebras, the algebraic analogues of \(n\)-homogeneous \(C^*\)-algebras.
MSC:
| 47L10 | Algebras of operators on Banach spaces and other topological linear spaces |
| 30H50 | Algebras of analytic functions of one complex variable |
Keywords:
subalgebras of \(C^*\)-algebras; matrix bundle; Riemann surface; \(C^*\)-envelope; Azumaya algebra; homogeneous \(C^*\)-algebraCitations:
Zbl 0321.47019References:
| [1] | Arveson, William B.: Subalgebras of \[C^{\ast } C\]*-algebras. Acta Math. 123, 141-224 (1969) · Zbl 0194.15701 |
| [2] | Hopenwasser, A.: Boundary representations on \[C^{\ast } C\]*-algebras with matrix units. Trans. Am. Math. Soc. 177, 483490 (1973) · Zbl 0264.46058 |
| [3] | Muhly, P.S., Solel, B.: Tensor algebras over \[C^*C\]∗-correspondences: representations, dilations, and \[C^*C\]∗-envelopes. J. Funct. Anal. 158(2), 389457 (1998) · Zbl 0912.46070 |
| [4] | Davidson, Kenneth R., Katsoulis, Elias G.: Dilating covariant representations of the non-commutative disc algebras. J. Funct. Anal. 259(4), 817831 (2010) · Zbl 1202.46079 |
| [5] | Dritschel, M.A., McCullough, S.A.: Boundary representations for families of representations of operator algebras and spaces. J. Oper. Theory 53(1), 159-167 (2005) · Zbl 1119.47311 |
| [6] | Matthew Kennedy and Orr Moshe Shalit: Essential normality, essential norms and hyperrigidity. J. Funct. Anal. 268(10), 2990-3016 (2015) · Zbl 1319.47003 |
| [7] | Katsoulis, E., Ramsey, C.: Crossed products of operator algebras. ArXiv e-prints, (2015) · Zbl 1475.47002 |
| [8] | Griesenauer, E., Muhly, P.S., Solel, B.: Boundaries, bundles and trace algebras. ArXiv e-prints, (2015) · Zbl 06982041 |
| [9] | Abrahamse, M.B., Douglas, R.G.: A class of subnormal operators related to multiply-connected domains. Adv. Math. 19(1), 106148 (1976) · Zbl 0321.47019 |
| [10] | Tomiyama, J., Takesaki, M.: Applications of fibre bundles to the certain class of \[C^{\ast } C\]*-algebras. Thoku Math. J. 13(2), 498-522 (1961) · Zbl 0113.09701 |
| [11] | Ahlfors, Lars V, Sario, Leo: Riemann Surfaces. Princeton Mathematical Series. No. 26. Princeton University Press, Princeton (1960) · Zbl 0196.33801 |
| [12] | Steenrod, N.: The Topology of Fibre Bundles. Princeton Mathematical Series, vol. 14. Princeton University Press, Princeton (1951) · Zbl 0054.07103 |
| [13] | Arveson, William: The noncommutative Choquet boundary. J. Am. Math. Soc. 21(4), 1065-1084 (2008) · Zbl 1207.46052 |
| [14] | Artin, M.: On Azumaya algebras and finite dimensional representations of rings. J. Algebra 11, 532-563 (1969) · Zbl 0222.16007 |
| [15] | Procesi, C.: On a theorem of M. Artin. J. Algebra 22, 309-315 (1972) · Zbl 0238.16015 |
| [16] | Massey, W.S.: Algebraic topology: An Introduction. Harcourt, Brace & World Inc, New York (1967) · Zbl 0153.24901 |
| [17] | Widom, \[H.: H_p\] Hp sections of vector bundles over Riemann surfaces. Ann. Math. 94(2), 304-324 (1971) · Zbl 0238.32014 |
| [18] | Katok, S.: Fuchsian Groups. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1992) · Zbl 0753.30001 |
| [19] | Gunning, R.C.: Lectures on Vector Bundles Over Riemann Surfaces. University of Tokyo Press, Princeton University Press, Tokyo, Princeton (1967) · Zbl 0163.31903 |
| [20] | Raeburn, I., Williams, D.P.: Morita Equivalence and Continuous-Trace \[C^*C\]∗-algebras, Volume 60 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (1998) · Zbl 0922.46050 |
| [21] | Grothendieck, A.: A general theory of fibre spaces with structure sheaf. In: National Science Foundation Research Project on Geometry of Function Space. University of Kansas, Dept. of Mathematics, (1955) |
| [22] | Gunning, R.C.: Lectures on Riemann Surfaces. Princeton Mathematical Notes. Princeton University Press, Princeton (1966) · Zbl 0175.36801 |
| [23] | Kobayashi, S.: Differential geometry of complex vector bundles, In: Volume 15 of Publications of the Mathematical Society of Japan. Princeton University Press, Princeton. Kanô Memorial Lectures, 5. (1987) · Zbl 0708.53002 |
| [24] | Grauert, H.: Seminars on analytic functions. Number v. 2 in Seminars on Analytic Functions. Institute for Advanced Study. (1958) · Zbl 0094.27201 |
| [25] | Rhrl, H.: Das Riemann-Hilbertsche problem der theorie der linearen differentialgleichungen. Math. Ann. 133, 125 (1957) · Zbl 0088.06001 |
| [26] | Dixmier, J.: \[C^*C\]∗-algebras. Translated from the French by Francis Jellett. North Holland Mathematical Library, vol. 15. North-Holland Publishing Co., Amsterdam-New York-Oxford (1977) · Zbl 0372.46058 |
| [27] | Davidson, Kenneth R., Kennedy, Matthew: The Choquet boundary of an operator system. Duke Math. J. 164(15), 2989-3004 (2015) · Zbl 1344.46041 |
| [28] | Bishop, Errett, de Leeuw, Karel: The representations of linear functionals by measures on sets of extreme points. Ann. Inst. Fourier Grenoble 9, 305-331 (1959) · Zbl 0096.08103 |
| [29] | Wermer, J.: Analytic disks in maximal ideal spaces. Am. J. Math. 86, 161-170 (1964) · Zbl 0128.11003 |
| [30] | Ahern, P.R., Sarason, Donald: The \[H^p\] Hp spaces of a class of function algebras. Acta Math. 117, 123-163 (1967) · Zbl 0146.37203 |
| [31] | Arveson, William: The noncommutative Choquet boundary II: hyperrigidity. Israel J. Math. 184, 349-385 (2011) · Zbl 1266.46045 |
| [32] | Kleski, C.: Boundary representations and pure completely positive maps. J. Oper. Theory 71(1), 4562 (2014) · Zbl 1349.46061 |
| [33] | Arens, Richard: The closed maximal ideals of algebras of functions holomorphic on a Riemann surface. Rend. Circ. Mat. Palermo 7, 245-260 (1958) · Zbl 0143.35904 |
| [34] | Formanek, E.: Central polynomials for matrix rings. J. Algebra 23, 129-132 (1972) · Zbl 0242.15004 |
| [35] | Razmyslov, J.P.: A certain problem of Kaplansky. Izv. Akad. Nauk SSSR Ser. Mat. 37, 483501 (1973) |
| [36] | DeMeyer, F., Ingraham, E.: Separable algebras over commutative rings. In: Lecture Notes in Mathematics, vol. 181. Springer, Berlin (1971) · Zbl 0215.36602 |
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