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analysis

Source: miklos schweitzer 1997 q8

September 27, 2021
Functional AnalysisHilbert Space

Problem Statement

Let H be an infinite dimensional, separable, complex Hilbert space and denote B(H)\cal B (\cal H) the H\cal H-algebra of its bounded linear operators. Consider the algebras l∞(N,B(H))=l_{\infty} ({\Bbb N}, \cal B (\cal H) ) = {(an)∣An∈B(H)\{ (a_n) | A_n \in\cal B (\cal H) (n∈N),sup⁡n∣∣An∣∣<∞}(n \in {\Bbb N}), \sup_n ||A_n|| <\infty \} C(βN,B(H))C(\beta {\Bbb N}, \cal B (\cal H) ) = {f:βN→B(H)∣\{ f: \beta {\Bbb N} \to \cal B (\cal H)| f is continuous }\} with pointwise operations and supremum norm. Show that these C*-algebras are not isometrically isomorphic. (Here, βN\beta {\Bbb N} denotes the Stone-Cech compactification of the set of natural numbers.)
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