# Duality for Actions and Coactions of Measured Groupoids on by Takehiko Yamanouchi

By Takehiko Yamanouchi

Via type of compact abelian team activities on semifinite injective components, Jones and Takesaki brought the proposal of an motion of a measured groupoid on a von Neumann algebra, which has confirmed to be an enormous device for this type of research. Elaborating in this suggestion, this paintings introduces a brand new notion of a measured groupoid motion which could healthy extra completely into the groupoid atmosphere. Yamanouchi additionally indicates the life of a canonical coproduct on each groupoid von Neumann algebra, which ends up in an idea of a coaction of a measured groupoid. Yamanouchi then proves duality among those gadgets, extending Nakagami-Takesaki duality for (co)actions of in the community compact teams on von Neumann algebras.

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**Example text**

6 there exists a nonsingular matrix L such that || A \\L < 1. Since || An \\L < || A \\nL it follows that || An \\L -> 0. 1 it follows that An —>0. , converges to A. If convergence holds, then we write A = A{1) + A™ + A™ + . . 4. The matrix I — B is nonsingular and the series I + B + B2 + . . converges if and only if S(B) < 1. 6) 36 2 / MATRIX PRELIMINARIES Proof. Let S{n) = I + B + B2 + · · + Bn. 2 it follows that || Bn+1 || -> 0. 6) holds. If on the other hand the series I + B + B2 + · · · converges, then one can easily show that Bn —► 0.

Evidently P can be joined to Q by a path consisting of line segments connecting properly adjacent mesh points. Somewhere along this path there must be a seg ment joining a point P' of S to a point P" of T. 17). Hence aitj φ 0 where i e S and y e T. This contradiction shows that T is empty and that A is irreducible. Next we show that A has weak diagonal dominance. 11). Moreover, since R is bounded, there exists points of Rh which are properly adjacent to one or more points of Sh. 13). Thus some of the di do not appear in the matrix A.

3. If A is an irreducible matrix with weak diagonal dom inance, then det A φ 0 and none of the diagonal elements of A vanishes. Proof. If i V = l , then αλ1 > 0 by weak diagonal dominance and det A φ 0. Suppose that N > 1 and aiti = 0 for some /. By the weak diagonal dominance we have aifj = 0 for all j . But this contradicts the irreducibility. For if S = {i} and T = W — {i}> we have aitj = 0 for all i eS,jeT. 4, if det A = 0, then there exists a solution w 9ε 0 of the homogeneous system Au = 0.