On neighborhoods of the Kuratowski imbedding beyond the first extremum of the diameter functional. (English) Zbl 0735.46049

Katz, Mikhail

Fundam. Math. 137, No. 3, 161-175 (1991).

For a metric space \((X,d)\) the Kuratowski isometrical imbedding \(X\ni x\mapsto d_ x\), where \(d_ x=d(x,y)\) for \(y\in X\), into the Banach algebra of all continuous real-valued functions \(f:X\to\mathbb{R}\) with the norm \(\| f\|=\sup\{| f(x)|;\;x\in X\}\) is considered. After defining the concept of a critical value of the diameter functional, \(\text{diam}_ X(Y)=\sup\{d(x,y);\;x,y\in Y\}\) for \(\emptyset\neq Y\subset X\), the homotopy type of \(\varepsilon\)-neighbourhoods of the circle \(S^ 1\) in corresponding Banach algebra where \(\lambda_ 1<2\varepsilon<\lambda_ 2\), \(\lambda_ 1\), \(\lambda_ 2\) being critical values of the diameter functional \(\text{diam}_{S^ 1}\), is determined. Next, a similar problem for the complex projective space is solved.

Reviewer: W.Waliszewski (Łódź)

MSC:

46M20	Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
46J10	Banach algebras of continuous functions, function algebras
54E35	Metric spaces, metrizability
53C70	Direct methods (\(G\)-spaces of Busemann, etc.)

Keywords:

Kuratowski isometrical imbedding; critical value of the diameter functional; homotopy type; Banach algebra; complex projective space