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The rectifiable metric on the space of projections in a \(C^*\)-algebra. (English) Zbl 0778.46039

In analogy to the concept of exponential length introduced by J. Ringrose [Proc. R. Soc. Edinb., Sect. A 121, No. 1/2, 55-71 (1922; review below)], the notion of projective length of a \(C^*\)-algebra \(A\) is defined to be the supremum over the rectifiable distances, in the space of projections of \(A\), of any two projections in the same path component. For a unital \(C^*\)-algebra one finds that projective length \(\leq\) exponential length. There is also a closely related notion of projective rank. The paper contains computations and estimates of the projective length and rank for several examples of \(C^*\)-algebras. The case of Banach algebras is also discussed briefly.

MSC:

46L05 General theory of \(C^*\)-algebras
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