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].

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].

Reviewer: Taduri S. Rao (Bangalore)