Contractive projections on 𝐶₀(𝐾)

Friedman, Yaakov; Russo, Bernard

doi:10.1090/S0002-9947-1982-0664029-4

Contractive projections on $C_{0}(K)$
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by Yaakov Friedman and Bernard Russo

Trans. Amer. Math. Soc. 273 (1982), 57-73

DOI: https://doi.org/10.1090/S0002-9947-1982-0664029-4

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Abstract:

We show that the range of a norm one projection on a commutative ${C^\ast }$-algebra has a ternary product structure (Theorem 2). We describe and characterize all such projections in terms of extreme points in the unit ball of the image of the dual (Theorem 1). We give necessary and sufficient conditions for the range to be isometric to a ${C^\ast }$-algebra (Theorem 4) and we show that the range is a ${C_\sigma }$-space (Theorem 5).

References

T. Andô, Contractive projections in $L_{p}$ spaces, Pacific J. Math. 17 (1966), 391–405. MR 192340, DOI 10.2140/pjm.1966.17.391
Jonathan Arazy and Yaakov Friedman, Contractive projections in $C_{1}$ and $C_{\infty }$, Mem. Amer. Math. Soc. 13 (1978), no. 200, iv+165. MR 481219, DOI 10.1090/memo/0200

The Banach space

Itzhak Bars and Murat Günaydin, Construction of Lie algebras and Lie superalgebras from ternary algebras, J. Math. Phys. 20 (1979), no. 9, 1977–1993. MR 546254, DOI 10.1063/1.524309
Ehrhard Behrends, $M$-structure and the Banach-Stone theorem, Lecture Notes in Mathematics, vol. 736, Springer, Berlin, 1979. MR 547509, DOI 10.1007/BFb0063153
S. J. Bernau and H. Elton Lacey, Bicontractive projections and reordering of $L_{p}$-spaces, Pacific J. Math. 69 (1977), no. 2, 291–302. MR 487416, DOI 10.2140/pjm.1977.69.291
Frank F. Bonsall and John Duncan, Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80, Springer-Verlag, New York-Heidelberg, 1973. MR 0423029, DOI 10.1007/978-3-642-65669-9
R. G. Douglas, Contractive projections on an ${\mathfrak {L}}_{1}$ space, Pacific J. Math. 15 (1965), 443–462. MR 187087, DOI 10.2140/pjm.1965.15.443
Edward G. Effros and Erling Størmer, Positive projections and Jordan structure in operator algebras, Math. Scand. 45 (1979), no. 1, 127–138. MR 567438, DOI 10.7146/math.scand.a-11830
I. Glicksberg, Some uncomplemented function algebras, Trans. Amer. Math. Soc. 111 (1964), 121–137. MR 161175, DOI 10.1090/S0002-9947-1964-0161175-8
Y. Friedman and B. Russo, Contractive projections on $C^{\ast }$-algebras, Operator algebras and applications, Part 2 (Kingston, Ont., 1980) Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 615–618. MR 679548
M. Günaydin, Quadratic Jordan formulation of quantum mechanics and construction of Lie (super) algebras from Jordan (super) algebras, Group theoretical methods in physics (Proc. Eighth Internat. Colloq., Kiryat Anavim, 1979) Ann. Israel Phys. Soc., vol. 3, Hilger, Bristol, 1980, pp. 279–296. MR 626844
Magnus R. Hestenes, A ternary algebra with applications to matrices and linear transformations, Arch. Rational Mech. Anal. 11 (1962), 138–194. MR 150166, DOI 10.1007/BF00253936
J. L. Kelley, Averaging operators on $C_{\infty }(X)$, Illinois J. Math. 2 (1958), 214–223. MR 103409, DOI 10.1215/ijm/1255381343
Gunnar Hans Olsen, On the classification of complex Lindenstrauss spaces, Math. Scand. 35 (1974), 237–258. MR 367626, DOI 10.7146/math.scand.a-11550
Walter Rudin, Real and complex analysis, 2nd ed., McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. MR 0344043
Daniel E. Wulbert, Some complemented function spaces in $C(X)$, Pacific J. Math. 24 (1968), 589–602. MR 223868, DOI 10.2140/pjm.1968.24.589
Daniel E. Wulbert, Averaging projections, Illinois J. Math. 13 (1969), 689–693. MR 256176
Daniel E. Wulbert, Convergence of operators and Korovkin’s theorem, J. Approximation Theory 1 (1968), 381–390. MR 235370, DOI 10.1016/0021-9045(68)90016-6
Joram Lindenstrauss and Daniel E. Wulbert, On the classification of the Banach spaces whose duals are $L_{1}$ spaces, J. Functional Analysis 4 (1969), 332–349. MR 0250033, DOI 10.1016/0022-1236(69)90003-2

Charakterisierung ternärer Operatorenringe

Robert E. Atalla, On the ergodic theory of contractions, Rev. Colombiana Mat. 10 (1976), no. 2, 75–81. MR 0417817