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Positive-definite kernels, length functions on groups and a noncommutative von Neumann inequality. (English) Zbl 0714.43007
Let L(H) denote the space of bounded operators on a Hilbert space H which is given the structure of a commutative Banach algebra by fixing a basis for H and multiplication by the Schur product $$(a*b)_{ij}=(a_{ij}b_{ij})$$ for the matrix representation of L(H). The basic results for complex-valued positive definite kernels also hold for operator-valued kernels, an example of which is the kernel $$(n,m)\mapsto A^{n-m}$$ and $$(n,m)\mapsto (A^*)^{m-n}$$ on $${\mathbb{Z}}$$, which is, by the way, considered as a metric space with the natural metric, and where A is a contraction on H. For sets $$S_ 1$$ and $$S_ 2$$ which intersect at a single point, the author constructs positive definite kernels on $$S_ 1\cup S_ 2$$ which are products of normalised positive definite kernels on $$S_ 1$$ and $$S_ 2$$ trivial for multiplication at the intersecting point. A similar construction defines an almost positive-definite (also called conditionally positive-definite) kernel which is a sum of almost positive-definite kernels trivial at the intersecting point. The author’s interest is in free-product groups and with sets which are metric spaces, so he works in the context of R-trees [see J. W. Morgan and P. B. Shalen, Ann. Math., II. Ser. 120, 401-476 (1984; Zbl 0583.57005), and J. P. Serre, Trees (1980; Zbl 0369.20013)], metric space generalisations of simply connected simplicial complexes. The main result is a criterion for positivity of kernels on R- trees, and he uses it to show that the metric of an $${\mathbb{R}}$$-tree is a negative-definite kernel [cf. for instance U. Haagerup, Invent. Math. 50, 279-293 (1979; Zbl 0408.46046)] and to generalise, to operator- valued kernels, his result [J. Reine Angew. Math. 377, 170-186 (1987; Zbl 0604.43004)] that a free product of positive kernels on a free product of discrete groups is positive. Another application is a generalisation of a much weakened form of a result of J. von Neumann [Math. Nachr. 4, 258-281 (1951; Zbl 0042.123)] to polynomials in noncommuting contractions $$A_ 1,A_ 2,...,A_ n$$ on H, showing that $$\| p(A_ 1,...,A_ n)\| \leq \sup \| p(U_ 1,...,U_ n)\|$$, where the norms are on L(H) and the supremum is over finite dimensional unitary operators on H.
Reviewer: A.Wulfsohn

##### MSC:
 43A35 Positive definite functions on groups, semigroups, etc. 20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 46J10 Banach algebras of continuous functions, function algebras
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