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Simultaneous upper triangular forms for commuting operators in a finite von Neumann algebra. (English) Zbl 1520.47079

Let \(\mathcal{M}\) be a von Neumann algebra equipped with a normal faithful tracial state. A well-known classical result is that commuting matrices can be simultaneously upper-triangularized. The authors establish analogous results for commuting families of elements of \(\mathcal{M}\). More precisely, by employing the results of H. Schultz [J. Funct. Anal. 236, No. 2, 457–489 (2006; Zbl 1101.47004)], they construct the joint Brown measure and joint Haagerup-Schultz projections, and then they relate them, in the case of a finite tuple \(T = (T_1, \dots,T_ n)\) of commuting operators, to various notions of joint spectrum (including the Taylor joint spectrum). They also use the joint Haagerup-Schultz subspaces to find simultaneous Schur-type upper triangular forms of tuples \(T\) of commuting operators of \(\mathcal{M}\). This is an extension of the main result of K. Dykema et al., [J. Reine Angew. Math. 708, 97–114 (2015; Zbl 1343.46056)], where the case of a single operator was treated. Furthermore, they consider the Arens multivariate holomorphic functional calculus applied to a commuting tuple \(T\). Finally, they prove that the joint spectral distribution measures and the joint Haagerup-Schultz projections behave well with respect to conjugation by invertible operators.

MSC:

47C15 Linear operators in \(C^*\)- or von Neumann algebras
47A60 Functional calculus for linear operators
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References:

[1] Albrecht, E., On joint spectra. Studia Math.64(1979), 263-271. . · Zbl 0412.47001
[2] Arens, R., The analytic-functional calculus in commutative topological algebras. Pacific J. Math.11(1961), 405-429. · Zbl 0109.34203
[3] Bourbaki, N., Éléments de mathématique. Fasc. XXXII. Théories spectrales. , Hermann, Paris, 1967. · Zbl 0152.32603
[4] Brown, L. G., Lidskii’s theorem in the type II case. In: Geometric methods in operator algebras (Kyoto, 1983). , Longman Sci. Tech., Harlow, 1986, pp. 1-35. · Zbl 0646.46058
[5] Dixmier, J., Von Neumann algebras. In: North-Holland Mathematical Library, 27. North-Holland Publishing, Amsterdam, 1981. · Zbl 0473.46040
[6] Dykema, K., Noles, J., Sukochev, F., and Zanin, D., On reduction theory and Brown measure for closed unbounded operators. J. Funct. Anal.371(2016), 3403-3422. · Zbl 1358.47026
[7] Dykema, K., Sukochev, F., and Zanin, D., A decomposition theorem in II_1-factors. J. Reine Angew. Math.708(2015), 97-114. . · Zbl 1343.46056
[8] Dykema, K., Sukochev, F., and Zanin, D., Holomorphic functional calculus on upper triangular forms in finite von Neumann algebras. Illinois J. Math.59(2015), 819-824. · Zbl 1353.47076
[9] Dykema, K., Sukochev, F., and Zanin, D., An upper triangular decomposition theorem for some unbounded operators affiliated to II_1-factors. Israel J. Math.222(2017), 645-709. · Zbl 1497.47056
[10] Fack, T. and Kosaki, H., Generalized s-numbers of 𝜏-measurable operators. Pacific J. Math.123(1986), 269-300. · Zbl 0617.46063
[11] Folland, G. B., Real analysis. Second ed., John Wiley & Sons, New York, 1999. · Zbl 0924.28001
[12] Haagerup, U. and Schultz, H., Brown measures of unbounded operators affiliated with a finite von Neumann algebra. Math. Scand.100(2007), 2, 209-263. · Zbl 1168.46039
[13] Haagerup, U. and Schultz, H., Invariant subspaces for operators in a general II_1-factor. Publ. Math. Inst. Hautes Études Sci.109(2009), 19-111. · Zbl 1178.46058
[14] Harte, R. E., Spectral mapping theorems. Proc. Roy. Irish Acad. Sect. A72(1972), 89-107. · Zbl 0206.13301
[15] Müller, V., On the Taylor functional calculus. Studia Math.150(2002), 79-97. · Zbl 1005.47017
[16] Putinar, M., Uniqueness of Taylor’s functional calculus. Proc. Am. Math. Soc.89(1983), 647-650. · Zbl 0573.47034
[17] Raeburn, I. and Sinclair, A. M., The C^∗-algebra generated by two projections. Math. Scand.65(1989), 278-290. . · Zbl 0717.46048
[18] Schultz, H., Brown measures of sets of commuting operators in a type II_1 factor. J. Funct. Anal.236(2006), 457-489. . · Zbl 1101.47004
[19] Taylor, J. L., The analytic-functional calculus for several commuting operators. Acta Math.125(1970), 1-38. . · Zbl 0233.47025
[20] Taylor, J. L., A joint spectrum for several commuting operators. J. Funct. Anal.6(1970), 172-191. · Zbl 0233.47024
[21] Vasilescu, F.-H., A characterization of the joint spectrum in Hilbert spaces. Rev. Roumaine Math. Pures Appl.22(1977), 1003-1009. · Zbl 0371.47035
[22] Vasilescu, F.-H., A Martinelli type formula for the analytic functional calculus. Rev. Roumaine Math. Pures Appl.23(1978), 1587-1605. · Zbl 0402.47011
[23] Vasilescu, F.-H., Analytic functional calculus and Martinelli’s formula. In: Romanian-Finnish Seminar on complex analysis. , Springer, Berlin, 1979, pp. 693-701. · Zbl 0425.47010
[24] Waelbrock, L., Le calcule symbolique dans les alg ‘ebres commutatives. J. Math. Pures Appl.33(1954), 147-186. · Zbl 0056.33601
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