The Curvature Invariant of a Non-commuting $N$-tuple

Details The Curvature Invariant of a Non-commuting $N$-tuple The Curvature Invariant of a Non-commuting $N$-tuple

2003-09-23

arxiv.org/abs/math/0309383v1

Integral Eqtns. & Operator Thy. 41 (2001), 426-454

Mathematics

Operator Algebras

29 pages, preprint version

Scientific paper

Non-commutative versions of Arveson's curvature invariant and Euler characteristic for a commuting $n$-tuple of operators are introduced. The non-commutative curvature invariant is sensitive enough to determine if an $n$-tuple is free. In general both invariants can be thought of as measuring the freeness or curvature of an $n$-tuple. The connection with dilation theory provides motivation and exhibits relationships between the invariants. A new class of examples is used to illustrate the differences encountered in the non-commutative setting and obtain information on the ranges of the invariants. The curvature invariant is also shown to be upper semi-continuous.

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