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Multiplicative integrals and geometry of spaces of projections. (English) Zbl 0756.46041

Let \(A\) be a Banach algebra with 1 (over \(\mathbb{R}\) or \(\mathbb{C}\)) and \[ Q_ n=\{q=(q_ 1,\dots,q_ n)\in A^ n:\;q_ k^ 2=q_ k,\;q_ i q_ k=0\text{ if } i\neq k,\;\sum q_ k^ 2=1\}. \] For \(q_ 0\in Q_ n\) the map \(\pi: G\to Q_ n\), \(\pi(g)=gq_ 0g^{-1}\) (\(G\) stands for the group of units of \(A\)) defines a principal fiber bundle over its image. For a “good” curve \(\gamma: [0,1]\to Q_ n\) with origin \(q_ 0\) the authors construct an explicit lift which turns out to be the horizontal lift of \(\gamma\) with respect to the natural connection. A similar result is given for the space \[ S_ q=\{(a,b)\in A^ 2:\;aq=a,\;qb=b,\;ba=q\}, \] where \(q\in A\) is a fixed idempotent. These constructions are based on the techniques of multiplicative integrals.

MSC:

46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
46H05 General theory of topological algebras
53C05 Connections (general theory)
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