Abstract
In earlier papers the second author and Charles Read have introduced and studied a new notion of positivity for operator algebras, with an eye to extending certain C*-algebraic results and theories to more general algebras. The present paper consists of complements to some facts in the just mentioned papers, concerning this notion of positivity. For example we prove a result on the numerical range of products of the roots of commuting operators with numerical range in a sector.
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Blecher D.P.: Noncommutative peak interpolation revisited. Bull. Lond. Math. Soc. 45, 1100–1106 (2013)
Blecher D.P., Hay D.M., Neal M.: Hereditary subalgebras of operator algebras. J. Oper. Theory 59, 333–357 (2008)
Blecher D.P., Le Merdy C.: Operator Algebras and their Modules—An Operator Space Approach. Oxford Univ. Press, Oxford (2004)
Blecher D.P., Neal M.: Metric characterizations of isometries and of unital operator spaces and systems. Proc. Am. Math. Soc. 139, 985–998 (2011)
Blecher D.P., Neal M.: Open projections in operator algebras I: comparison theory. Studia Math. 208, 117–150 (2012)
Blecher D.P., Neal M.: Open projections in operator algebras II: compact projections. Studia Math. 209, 203–224 (2012)
Blecher D.P., Read C.J.: Operator algebras with contractive approximate identities. J. Funct. Anal. 261, 188–217 (2011)
Blecher D.P., Read C.J.: Operator algebras with contractive approximate identities II. J. Funct. Anal. 204, 1049–1067 (2013)
Blecher, D.P., Read, C.J.: Operator algebras with contractive approximate identities III (preprint) (2013). arXiv:1308.2723v2
Gustafson, K.E., Rao, D.K.M.: Numerical range. The field of values of linear operators and matrices. Universitext. Springer, New York (1997)
Haase, M.: The functional calculus for sectorial operators. Operator Theory: Advances and Applications, vol. 169, Birkhauser Verlag, Basel (2006)
Horn R.A., Johnson C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1994)
Lancien F., Lancien G., Le Merdy C.: A joint functional calculus for sectorial operators with commuting resolvents. Proc. Lond. Math. Soc. 77, 387–414 (1998)
Li C-K., Rodman L., Spitkovsky I.M.: On numerical ranges and roots. J. Math. Anal. Appl. 282, 329–340 (2003)
Macaev V.I., Palant J.A.: On the powers of a bounded dissipative operator (Russian). Ukrain. Mat. Z. 14, 329–337 (1962)
Paulsen V.I.: Completely bounded maps and operator algebras. Cambridge Studies in Advanced Mathematics, vol. 78. Cambridge University Press, Cambridge (2002)
Pedersen G.K.: C*-Algebras and their Automorphism Groups. Academic Press, London (1979)
Read C.J.: On the quest for positivity in operator algebras. J. Math. Anal. Appl. 381, 202–214 (2011)
Sz.-Nagy, B., Foias, C., Bercovici, H., Kerchy, L.: Harmonic Analysis of Operators on Hilbert Space, 2nd edn. Universitext. Springer, New York (2010)
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C. A. Bearden and D. P. Blecher were supported by a grant from the NSF.
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Bearden, C.A., Blecher, D.P. & Sharma, S. On Positivity and Roots in Operator Algebras. Integr. Equ. Oper. Theory 79, 555–566 (2014). https://doi.org/10.1007/s00020-014-2136-y
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DOI: https://doi.org/10.1007/s00020-014-2136-y