article · Proceedings of the American Mathematical Society · 1984

Completely bounded homomorphisms of operator algebras

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Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a unital operator algebra. We prove that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho"> <mml:semantics> <mml:mi>ρ</mml:mi> <mml:annotation encoding="application/x-tex">\rho</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a completely bounded, unital homomorphism of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> into the algebra of bounded operators on a Hilbert space, then there exists a similarity <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper S Superscript negative 1 Baseline double-vertical-bar dot double-vertical-bar upper S double-vertical-bar equals double-vertical-bar rho double-vertical-bar Subscript c b"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mo symmetric="true">‖</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:mrow> <mml:mo symmetric="true">‖</mml:mo> </mml:mrow> <mml:mo>⋅</mml:mo> <mml:mrow> <mml:mo symmetric="true">‖</mml:mo> <mml:mi>S</mml:mi> <mml:mo symmetric="true">‖</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow> <mml:mo symmetric="true">‖</mml:mo> <mml:mi>ρ</mml:mi> <mml:mo symmetric="true">‖</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>c</mml:mi> <mml:mi>b</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\left \| {{S^{ - 1}}} \right \| \cdot \left \| S \right \| = {\left \| \rho \right \|_{cb}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Superscript negative 1 Baseline rho left-parenthesis dot right-parenthesis upper S"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:mi>ρ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mi>S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{S^{ - 1}}\rho ( \cdot )S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a completely contractive homomorphism. We also show how Rota’s theorem on operators similar to contractions and the result of Sz.-Nagy and Foias on the similarity of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho"> <mml:semantics> <mml:mi>ρ</mml:mi> <mml:annotation encoding="application/x-tex">\rho</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dilations to contractions can be deduced from this result.

Cite this paper

APA 7

Paulsen, V. I. (1984). Completely bounded homomorphisms of operator algebras. Proceedings of the American Mathematical Society, 92(2), 225–228. https://doi.org/10.1090/s0002-9939-1984-0754708-x

MLA 9

Paulsen, Vern I. “Completely bounded homomorphisms of operator algebras.” Proceedings of the American Mathematical Society, vol. 92, no. 2, 1984, pp. 225–28, https://doi.org/10.1090/s0002-9939-1984-0754708-x.

Chicago

Paulsen, Vern I. 1984. “Completely bounded homomorphisms of operator algebras.” Proceedings of the American Mathematical Society 92 (2): 225–28. https://doi.org/10.1090/s0002-9939-1984-0754708-x.

Harvard

Paulsen, V.I. (1984) ‘Completely bounded homomorphisms of operator algebras’, Proceedings of the American Mathematical Society, 92(2), pp. 225–228. Available at: https://doi.org/10.1090/s0002-9939-1984-0754708-x.

BibTeX

@article{paulsen1984completely,
  title = {{Completely bounded homomorphisms of operator algebras}},
  author = {Paulsen, Vern I.},
  journal = {Proceedings of the American Mathematical Society},
  volume = {92},
  number = {2},
  pages = {225--228},
  year = {1984},
  doi = {10.1090/s0002-9939-1984-0754708-x}
}

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