Banach Journal of Mathematical Analysis

On domains of unbounded derivations of generalized B-algebras

Martin Weigt and Ioannis Zarakas

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We study properties under which the domain of a closed derivation δ:D(δ)A of a generalized B-algebra A remains invariant under analytic functional calculus. For a complete, generalized B-algebra with jointly continuous multiplication, two sufficient conditions are assumed: that the unit of A belongs to the domain of the derivation, along with a condition related to the coincidence σA(x)=σD(δ)(x) of the (Allan) spectra for every element xD(δ). Certain results are derived concerning the spectra for a general element of the domain, in the realm of a domain which is advertibly complete or enjoys the Q-property. For a closed -derivation δ of a complete GB-algebra with jointly continuous multiplication such that 1D(δ) and x a normal element of the domain, f(x)D(δ) for every analytic function on a neighborhood of the spectrum of x. We also give an example of a closed derivation of a GB-algebra which does not contain the identity element. A condition for a closed derivation of a GB-algebra A to be the generator of a one-parameter group of automorphisms of A is provided along with a generalization of the Lumer–Phillips theorem for complete locally convex spaces.

Article information

Banach J. Math. Anal., Volume 12, Number 4 (2018), 873-908.

Received: 17 May 2017
Accepted: 1 November 2017
First available in Project Euclid: 20 April 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46H05: General theory of topological algebras
Secondary: 46H35: Topological algebras of operators [See mainly 47Lxx] 46K05: General theory of topological algebras with involution 46L05: General theory of $C^*$-algebras

$\mathrm{GB}^{*}$-algebra topological algebra derivation


Weigt, Martin; Zarakas, Ioannis. On domains of unbounded derivations of generalized B $^{*}$ -algebras. Banach J. Math. Anal. 12 (2018), no. 4, 873--908. doi:10.1215/17358787-2017-0060.

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  • [1] G. R. Allan, A spectral theory for locally convex algebras, Proc. Lond. Math. Soc. (3) 15 (1965), 399–421.
  • [2] G. R. Allan, On a class of locally convex algebras, Proc. Lond. Math. Soc. (3) 17 (1967), 91–114.
  • [3] V. A. Babalola, Semigroups of operators on locally convex spaces, Trans. Amer. Math. Soc. 199 (1974), 163–179.
  • [4] F. Bagarello, M. Fragoulopoulou, A. Inoue, and C. Trapani, The completion of a $C^{*}$-algebra with a locally convex topology, J. Operator Theory 56 (2006), no. 2, 357–376.
  • [5] F. Bagarello, A. Inoue, and C. Trapani, Derivations of quasi $^{*}$-algebras, Int. J. Math. Math. Sci. 2004, no. 21–24, 1077–1096.
  • [6] S. J. Bhatt, A note on generalized $B^{*}$-algebras, J. Indian Math. Soc. (N.S.) 43 (1979), no. 1–4, 253–257.
  • [7] O. Bratteli, Derivations, Dissipations and Group Actions of $C^{*}$-Algebras, Lecture Notes in Math. 1229, Springer, Berlin, 1986.
  • [8] O. Bratteli and D. W. Robinson, Unbounded derivations of C$^{*}$-algebras, Comm. Math. Phys. 42 (1975), 253–268.
  • [9] P. G. Dixon, Generalized $B^{*}$-algebras, Proc. Lond. Math. Soc. (3) 21 (1970), 693–715.
  • [10] M. Fragoulopoulou, Topological Algebras with Involution, North-Holland Math. Stud. 200, North-Holland, Amsterdam, 2005.
  • [11] M. Fragoulopoulou, A. Inoue, and K.-D. Kürsten, “Old and new results on Allan’s $\mathrm{GB}^{*}$-algebras” in Banach Algebras 2009, Banach Center Publ. 91, Polish Acad. Sci. Inst. Math., Warsaw, 2010, 169–178.
  • [12] M. Fragoulopoulou, A. Inoue, and M. Weigt, Tensor products of generalized $B^{*}$-algebras, J. Math. Anal. Appl. 420 (2014), no. 2, 1787–1802.
  • [13] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, Amer. Math. Soc. Colloq. Publ. 31, Amer. Math. Soc., Providence, 1957.
  • [14] A. Inoue and K.-D. Kürsten, On $C^{*}$-like locally convex $*$-algebras, Math. Nachr. 235 (2002), 51–58.
  • [15] E. Kissin, Totally symmetric algebras and the similarity problem, J. Funct. Anal. 77 (1988), no. 1, 88–97.
  • [16] E. Kissin and V. S. Shulman, Dense $Q$-subalgebras of Banach and $C^{*}$-algebras and unbounded derivations of Banach and $C^{*}$-algebras, Proc. Edinb. Math. Soc. (2) 36 (1993), no. 2, 261–276.
  • [17] G. Kothe, Topological Vector Spaces, II, Grundlehren Math. Wiss. 237, Springer, New York, 1979.
  • [18] W. Kunze, Zur algebraischen Struktur der ${GC}^{*}$-Algebren, Math. Nachr. 88 (1979), 7–11.
  • [19] G. Lumer and R. S. Phillips, Dissipative operators in a Banach space, Pacific J. Math. 11 (1961), 679–698.
  • [20] A. Mallios, Hermitian $K$-theory over topological $^{*}$-algebras, J. Math. Anal. Appl. 106 (1985), no. 2, 454–539.
  • [21] A. Y. Pirkovskii, Arens–Michael envelopes, homological epimorphisms, and relatively quasifree algebras (in Russian), Tr. Mosk. Mat. Obs. 69 (2008), 34–125; English translation in Trans. Moscow Math. Soc. 2008, 27–104.
  • [22] S. Sakai, On one-parameter subgroups of $^{*}$-automorphisms on operator algebras and the corresponding unbounded derivations, Amer. J. Math. 98 (1976), no. 2, 427–440.
  • [23] S. Sakai, Operator Algebras in Dynamical Systems: The Theory of Unbounded Derivations in $C^{*}$-Algebras, Encyclopedia Math. Appl. 41, Cambridge Univ. Press, Cambridge, 1991.
  • [24] K. Schmüdgen, Unbounded Operator Algebras and Representation Theory, Oper. Theory Adv. Appl. 37, Birkhäuser, Basel, 1990.
  • [25] M. Takesaki, Theory of Operator Algebras, I, Springer, New York, 1979.
  • [26] A. E. Taylor, Spectral theory of closed distributive operators, Acta Math. 84 (1951), 189–224.
  • [27] I. D. Tembo, “Invertibility in the algebra of $\tau$-measurable operators” in Operator Algebras, Operator Theory and Applications, Oper. Theory Adv. Appl. 195, Birkhäuser, Basel, 2010, 245–256.
  • [28] L. Waelbroeck, Topological Vector Spaces and Algebras, Lecture Notes in Math. 230, Springer, Berlin, 1971.
  • [29] S. Warner, Inductive limits of normed algebras, Trans. Amer. Math. Soc. 82 (1956), 190–216.
  • [30] M. Weigt and I. Zarakas, Derivations of Fréchet nuclear $\mathrm{GB}^{*}$-algebras, Bull. Aust. Math. Soc. 92 (2015), no. 2, 290–301.
  • [31] M. Weigt and I. Zarakas, Spatiality of derivations of Fréchet $\mathrm{GB}^{*}$-algebras, Studia Math. 231 (2015), no. 3, 219–239.
  • [32] K. Yosida, Functional Analysis, 5th ed., Grundlehren Math. Wiss. 123, Springer, Berlin, 1978.
  • [33] W. Zelazko, Metric Generalizations of Banach Algebras, Rozprawy Mat. 47, Warsaw, 1965.
  • [34] W. Zelazko, On maximal ideals in commutative $m$-convex algebras, Studia Math. 58 (1976), no. 3, 291–298.