Averaging operators in and lower semicontinuous sections of continuous maps

Author:
Seymour Z. Ditor

Journal:
Trans. Amer. Math. Soc. **175** (1973), 195-208

MSC:
Primary 46E15; Secondary 46J10, 47B99

DOI:
https://doi.org/10.1090/S0002-9947-1973-0312228-8

MathSciNet review:
0312228

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Abstract | References | Similar Articles | Additional Information

Abstract: For certain kinds of compact Hausdorff spaces *S*, necessary and sufficient topological conditions are provided for determining if there exists a norm 1 projection of onto any given separable selfadjoint subalgebra *A*, the conditions being in terms of the decomposition that *A* induces on *S*. In addition, for arbitrary *S* and selfadjoint closed subalgebra *A* of , some results on lower bounds for the norms of projections of onto *A* are obtained. An example is given which shows that the greatest lower bound of the projection norms need not be attained.

**[1]**D. Amir,*Projections onto continuous function spaces*, Proc. Amer. Math. Soc.**15**(1964), 396-402. MR**29**#2634. MR**0165350 (29:2634)****[2]**R. Arens,*Projections on continuous function spaces*, Duke Math. J.**32**(1965), 469-478. MR**31**#6108. MR**0181882 (31:6108)****[3]**H. H. Corson and J. Lindenstrauss,*Continuous selections with nonmetrizable range*, Trans. Amer. Math. Soc.**121**(1966), 492-504. MR**32**#4667. MR**0187214 (32:4667)****[4]**S. Ditor,*On a lemma of Milutin concerning averaging operators in continuous function spaces*, Trans. Amer. Math. Soc.**149**(1970), 443-452. MR**0435921 (55:8872)****[5]**P. R. Halmos,*Measure theory*, Van Nostrand, Princeton, N.J., 1950. MR**11**, 504. MR**0033869 (11:504d)****[6]**E. A. Michael,*Topologies on spaces of subsets*, Trans. Amer. Math. Soc.**71**(1951), 151-182. MR**13**, 54. MR**0042109 (13:54f)****[7]**-,*Selected selection theorems*, Amer. Math. Monthly**63**(1956), 233-238. MR**1529282****[8]**-,*A linear mapping between function spaces*, Proc. Amer. Math. Soc.**15**(1964), 407-409. MR**28**#5327. MR**0162128 (28:5327)****[9]**A. Pełczyński,*Linear extensions, linear averagings, and their applications to linear topological classification of spaces of continuous functions*, Dissertationes Math. Rozprawy Mat.**58**(1968), 92 pp. MR**37**#3335. MR**0227751 (37:3335)****[10]**K. A. Ross and A. H. Stone,*Products of separable spaces*, Amer. Math. Monthly**71**(1964), 398-403. MR**29**#1611. MR**0164314 (29:1611)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1973-0312228-8

Keywords:
Averaging operator,
regular averaging operator,
projection operator,
complemented subalgebra,
continuous function space,
Milutin space,
lower semicontinuous section,
lower bounds for norms of averaging operators

Article copyright:
© Copyright 1973
American Mathematical Society