## Completely bounded linear extensions of operator-valued functions on $^ *$-semigroups

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- by Ching Yun Suen
- Proc. Amer. Math. Soc.
**105**(1989), 330-334 - DOI: https://doi.org/10.1090/S0002-9939-1989-0931737-9
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## Abstract:

Let $G$ be a unital $*$-semigroup [7, p. 1] in a unital (complex) ${C^*}$-algebra such that the linear span of $G$ is norm dense in it. Extending the results of [6], we have completely bounded linear extension theorems of operatorvalued functions on $G$. Applying extension theorems, we have that each regular bounded operator measure has the form $V_1^*F(){V_2}$, where ${V_1}$ and ${V_2}$ are linear operators and $F$ is a selfadjoint spectral operator measure.## References

- William B. Arveson,
*Subalgebras of $C^{\ast }$-algebras*, Acta Math.**123**(1969), 141–224. MR**253059**, DOI 10.1007/BF02392388 - D. W. Hadwin,
*Dilations and Hahn decompositions for linear maps*, Canadian J. Math.**33**(1981), no. 4, 826–839. MR**634141**, DOI 10.4153/CJM-1981-064-7 - Vern I. Paulsen,
*Every completely polynomially bounded operator is similar to a contraction*, J. Funct. Anal.**55**(1984), no. 1, 1–17. MR**733029**, DOI 10.1016/0022-1236(84)90014-4 - Vern I. Paulsen,
*Completely bounded maps and dilations*, Pitman Research Notes in Mathematics Series, vol. 146, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1986. MR**868472** - Vern I. Paulsen and Ching Yun Suen,
*Commutant representations of completely bounded maps*, J. Operator Theory**13**(1985), no. 1, 87–101. MR**768304** - Z. Sebestyén,
*Dilatable operator valued functions on $C^{\ast }$-algebras*, Acta Math. Hungar.**43**(1984), no. 1-2, 37–42. MR**731960**, DOI 10.1007/BF01951322 - F. H. Szafraniec,
*Dilations on involution semigroups*, Proc. Amer. Math. Soc.**66**(1977), no. 1, 30–32. MR**473873**, DOI 10.1090/S0002-9939-1977-0473873-1 - W. Forrest Stinespring,
*Positive functions on $C^*$-algebras*, Proc. Amer. Math. Soc.**6**(1955), 211–216. MR**69403**, DOI 10.1090/S0002-9939-1955-0069403-4
B. Sz.-Nagy, - Gerd Wittstock,
*Ein operatorwertiger Hahn-Banach Satz*, J. Functional Analysis**40**(1981), no. 2, 127–150 (German, with English summary). MR**609438**, DOI 10.1016/0022-1236(81)90064-1

*Extensions of linear transformations in Hilbert space which extend beyond this space*, Appendix to

*Functional Analysis*(F. Riesz, B. Sz.-Nagy, coauthors), Ungar, New York, 1960.

## Bibliographic Information

- © Copyright 1989 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**105**(1989), 330-334 - MSC: Primary 46L05; Secondary 47A20, 47D99
- DOI: https://doi.org/10.1090/S0002-9939-1989-0931737-9
- MathSciNet review: 931737