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Means and convex combinations of unitary operators. (English) Zbl 0573.46034

The ways in which an element of the unit ball of a \(C^*\)-algebra \({\mathfrak A}\) can be expressed as convex combinations of elements of \({\mathcal U}({\mathfrak A})\), the group of unitary operators in \({\mathfrak A}\), is studied. It is shown that if S in \({\mathfrak A}\) has norm less than \(1- 2n^{-1}\), then S is the mean of n unitary operators in \({\mathfrak A}\). By examples, this result is shown to be ”sharp.” The Murray-von Neumann result stating that a self-adjoint A in the closed unit ball of \({\mathfrak A}\) is a mean of two unitary operators is extended to describe exactly which such A can be expressed as \(aU+(1-a)V\) for a given a in [0,1/2] and U,V in \({\mathfrak U}({\mathfrak A})\). This decomposition is possible exactly when A has no spectrum in (2a-1,\(1-2a)\) and \(\| A\| \leq 1\). This last result and a detailed analytic-combinatorial analysis of the permutation polytope P generated by a point \((a_ 1,...,a_ n)\) in \({\mathbb{R}}^ n\), such that \(\sum a_ j=1\) and each \(a_ j\geq 0\), are used to show that if \(S=a_ 1U_ 1+...+a_ nU_ n\) \((U_ j\in {\mathcal U}({\mathfrak A}))\), then \(S=b_ 1V_ 1+...+b_ nV_ n\) for some \(V_ j\) in \({\mathcal U}({\mathfrak A})\) provided \((b_ 1,...,b_ n)\in P\). In particular \(S=n^{-1}(V_ 1+...+V_ n)\) for some \(V_ j\) in \({\mathcal U}({\mathfrak A})\). Conditions on the ”asymmetry” of the coefficients \(a_ 1,...,a_ n\) are established if \(\sum a_ jU_ j\) is a convex decomposition of S and no such decomposition with fewer unitary operators is possible.
These results are used to give another proof [cf. J. Glimm - R. Kadison, Pac. J. Math. 10, 547-556 (1960; Zbl 0152.330)] of the Gelfand-Neumark conjecture that the adjoint operation is isometric in a \(B^*\)-algebra. They represent a simplification and an extensive refinement of the Russo-Dye theorem. They also provide an attractive possibility for attacking the problem of ”non-commutative topological dimension.”

MSC:

46L10 General theory of von Neumann algebras
47L07 Convex sets and cones of operators
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)

Citations:

Zbl 0152.330
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