On Conditional Expectations Arising from Group Actions

Induced conditional expectations of nite index on crossed product C algebras are considered which are non algebraically of nite index The characteristics of actions of amenable topological groups on compact Hausdor spaces X are investigated ensuring the appearance of a well de ned induced conditional expectation on the corresponding commutative C algebra C X and its property to be of nite index For this purpose Hilbert C module and topological techniques are used Special emphasis is placed on discrete group actions In the light of recent results by E Kirchberg and the rst author we generalize some interesting results of Mahmood Khoshkam concerning special conditional ex pectations on crossed product C algebras which are of nite index in the sense of Y Watatani s algebraic de nition to the broader setting of similar conditional ex pectations E on them which are of nite index in the sense of M Baillet Y Denizeau and J F Havet i e those for which there exists a nite real number K so that the mapping K E id is positive Subsequently we consider actions of amenable in nite discrete or topological groups on locally compact Hausdor spaces which give rise to conditional expectations of nite index In these cases the action of an amenable topological group G can always be replaced by a suitable action of the completely disconnected group arising as the factor group of G by its connected component of the identity The in uence of the maximal length of orbit in X under the group action on the characteristic constant K EG of the derived conditional expectation of nite index on C X is described In addition we analyze the constructions of conditional expectations of nite index on commutative C algebras C X given by Y Watatani Propositions and which arise from actions of nite groups on compact Hausdor spaces X Special attention is paid to the close interrelation between non maximal orbits in X under the group action and of the non algebraic character of the corresponding conditional expectation of nite index For these investigations we use group action and Hilbert C module techniques Examples are used to illustrate the phenomena appearing for discrete group actions We make use of some basic notions and results from the theory of Hilbert C modules during our considerations to streamline the explanations For an introduction M Frank University of Leipzig Inst Math D Leipzig e mail frank mathematik uni leipzig de V M Manuilov Moscow State University Dept Mech Math Moscow Russia e mail troitsky mech math msu su E V Troitsky Moscow State University Dept Mech Math Moscow Russia e mail manuilov mech math msu su to this theory we refer the reader to the papers by W L Paschke and M A Rie el On liftings of conditional expectations of nite index to crossed products Throughout the present section we only deal with discrete groups G If a group G acts on a C algebra A as a group of automorphisms we denote the full and the reduced crossed product C algebras by A G and A r G respectively cf We consider conditional expectations E A B A on C algebras A i e projections of norm one with range C subalgebra B A see for detailed results The C algebra A can always be assumed to be unital without loss of generality since E can be continued to a normal conditional expectation E on the bidual W algebra A of A with range C algebra E A A which is isomorphic to the bidual W algebra B of B whereby the reduction of E to the canonical embedding A A recovers E A conditional expectation E is said to be faithful if E x x implies x for x A In this case both A and E A B share a common identity To consider both conditional expectations E A B A and actions of groups G on the C algebras involved we suppose that the group action of G by automorphisms on A commutes with the conditional expectation E We shall investigate conditional expectations of nite index both in the sense of Y Watatani and in the sense of M Baillet Y Denizeau and J F Havet If we considerW factors and normal conditional expectations on them with factor imageW algebra both notions reduce to the classical index notion of V Jones M Pimsner and S Popa A conditional expectation E A B A is said to be algebraically of nite index if there exist elements fu ung A a quasi basis such that the equality x Pn i uiE u i x is valid for every x A In this case the index is de ned by Ind E Pn i uiu i which is a positive invertible element in the center of A such that Ind E A and it does not depend on the choice of the quasi basis fu ung inside A and in particular not on the number of its elements From another point of view the existence of a nite quasi basis for some conditional expectation E A B A is equivalent to the algebraic statement that A is a nitely generated projective or equivalently Hilbert B module There is another interesting class of conditional expectations E A B A those for which there exists a nite real number K such that the mapping K E idA is a positive map on A They have the following remarkable property Proposition Let A be a C algebra and E A B A be a conditional expectation with xed point set B There then exists a nite real number K such that the mapping K E idA is positive if and only if E is faithful and the right pre Hilbert B module fA E h iA g is complete with respect to the norm kE h iA k B where ha biA a b for a b A Proof If A is complete with respect to the norm kE h iA k B then by the general theory of Banach spaces there exists a number K such that KkE x x k kx xk holds for every x A Set x a E a a for a A and and observe that