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.”


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


Zbl 0152.330
Full Text: DOI EuDML