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The Banach spaces $$C(K)$$. (English) Zbl 1040.46018
Johnson, W. B. (ed.) et al., Handbook of the geometry of Banach spaces. Volume 2. Amsterdam: North-Holland (ISBN 0-444-51305-1/hbk). 1547-1602 (2003).
In this ‘State of the Art’ article from the Handbook of Geometry of Banach spaces, the author describes the developments in the isomorphic theory of the Banach space $$C(K)$$ when $$K$$ is an infinite compact metric space.
Chapter II deals with the isomorphic classification of $$C(K)$$ spaces. Starting with the theorem of A. A. Milyutin [Teor. Funkts., Funkts. Anal. Prilozh. 2, 150–156 (1966; Zbl 0253.46050)] which says that for an uncountable compact metric space $$K$$, $$C(K)$$ is isomorphic to $$C([0,1])$$, this chapter covers the results of C. Bessaga and A. Pełczyński [Stud. Math. 19, 53–62 (1960; Zbl 0094.30303)] on isomorphic classification of $$C(K)$$ when $$K$$ is countable and the work of C. Samuel [Publ. Math. Univ. Paris VII, 81-91 (1983; Zbl 0589.93020)] involving the Szlenk index.
Some properties like weak injectivity, $$c_0$$-saturation and uncomplemented embeddings of $$C([0,1])$$ and $$C(\omega^{\omega})$$ in themselves are considered in Chapter III.
Chapter IV deals with operators on $$C(K)$$ spaces. Main emphasis here is on operators that fix copies of $$c_0$$, $$C([0,1])$$ and $$C(\omega^{\omega})$$. For example a theorem due to the author [Isr. J. Math. 13, 361–378 (1972; Zbl 0253.46048)] says that an operator $$T :C(K) \rightarrow X$$ such that $$T^\ast(X^\ast)$$ is non-separable fixes a copy of $$C([0,1])$$.
The final chapter of the article discusses the complemented subspace problem ($$CSP$$), ‘Is every infinite dimensional, complemented subspace $$X \subset C(K)$$, isomorphic to $$C(L)$$ for some compact Hausdorff space $$L$$?’ Starting with the result of A. Pełczyński, [Bull. Acad. Pol. Sci., Math. Astron. Phys. 10, 265–270 (1962; Zbl 0107.32504)] that any such $$X$$ has a copy of $$c_0$$, this chapter also deals with a dichotomy due to Y. Benyamini [Isr. J. Math. 29, 24–30 (1978; Zbl 0367.46014)] that says that such an $$X$$ is isomorphic to $$c_0$$ or $$C(\omega^{\omega})$$ embeds into $$X$$. Some problems mentioned here are: is a non-separable $$C(K)$$ isomorphic to a $$C(L)$$ for a totally disconnected $$L$$?, or is an infinite dimensional complemented subspace of $$C(K)$$ that contains a reflexive subspace isomorphic to $$C([0,1])$$?
Quoting from the Introduction: “An exciting new research development deals with many of the issues discussed here in the context of $$C^\ast$$-algebras. We shall only briefly allude to two discoveries.
The first is Kirchberg’s non-commutative analogue of Milutin’s theorem: every separable non-type $$I$$ nuclear $$C^\ast$$-algebra is completely isomorphic to the CAR algebra [E. Kirchberg, J. Funct. Anal. 129, 35–63 (1995; Zbl 0912.46059)].
The second concerns the quantized formulation of the separable extension property, due to the author [J. Oper. Theory 43, 329–374 (2000; Zbl 0992.47035)], and the joint theorem of Oikhberg and the author: the space of compact operators on a separable Hilbert space has the complete Separable Complementation Property [T. Oikhberg and H. P. Rosenthal, J. Funct. Anal. 179, 251–308 (2001; Zbl 1064.47072)].
For a recent survey and perspective on these developments, see H. P. Rosenthal [Contemp. Math. 321, 275–294 (2003; Zbl 1041.46040)].”
For the entire collection see [Zbl 1013.46001].

##### MSC:
 46B25 Classical Banach spaces in the general theory 46E15 Banach spaces of continuous, differentiable or analytic functions 46-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to functional analysis
##### Keywords:
space of continuous functions; isomorphisms; operators