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FIXED POINT ALGEBRAS
"... Although selfreference in arithmetic was used to impressive effect by Gödel in 1930 (published in 1931) when he noted the sentence asserting its own unprovability to be unprovable, and although this use immediately appealed to philosophers and philosophical logicians, it has largely been ignored by ..."
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to this success. The central notion of this exposition is that of a fixed point algebra. This notion is a new one—it is untested and, hence, of only provisional interest. But it does appear useful:
GENERALIZED FIXED POINT ALGEBRAS AND SQUAREINTEGRABLE GROUP ACTIONS
, 2000
"... Abstract. We analzye Rieffel’s construction of generalized fixed point algebras in the setting of group actions on Hilbert modules. Let G be a locally compact group acting on a C ∗algebra B. We construct a Hilbert module F over the reduced crossed product of G and B, using a pair (E, R), where E is ..."
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Cited by 8 (1 self)
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Abstract. We analzye Rieffel’s construction of generalized fixed point algebras in the setting of group actions on Hilbert modules. Let G be a locally compact group acting on a C ∗algebra B. We construct a Hilbert module F over the reduced crossed product of G and B, using a pair (E, R), where E
Permanence properties for crossed products and fixed point algebras of finite groups
 Trans. Amer. Math. Soc
"... Abstract. Let α: G → Aut(A) be an action of a finite group G on a C*algebra A. We present some conditions under which properties of A pass to the crossed product C∗(G,A, α) or the fixed point algebra Aα. We mostly consider the ideal property, the projection property, topological dimension zero, and ..."
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Cited by 4 (1 self)
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Abstract. Let α: G → Aut(A) be an action of a finite group G on a C*algebra A. We present some conditions under which properties of A pass to the crossed product C∗(G,A, α) or the fixed point algebra Aα. We mostly consider the ideal property, the projection property, topological dimension zero
Functoriality of Rieffel’s Generalised FixedPoint Algebras for Proper Actions
, 909
"... Abstract. We consider two categories of C ∗algebras; in the first, the isomorphisms are ordinary isomorphisms, and in the second, the isomorphisms are Morita equivalences. We show how these two categories, and categories of dynamical systems based on them, crop up in a variety of C ∗algebraic cont ..."
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contexts. We show that Rieffel’s construction of a fixedpoint algebra for a proper action can be made into functors defined on these categories, and that his Morita equivalence then gives a natural isomorphism between these functors and crossedproduct functors. These results have interesting applications
FIXEDPOINT ALGEBRAS FOR PROPER ACTIONS AND CROSSED PRODUCTS BY HOMOGENEOUS SPACES
, 907
"... Abstract. We consider a fixed free and proper action of a locally compact group G on a space T, and actions α: G → Aut A on C ∗algebras for which there is an equivariant embedding of (C0(T), rt) in (M(A), α). A recent theorem of Rieffel implies that α is proper and saturated with respect to the sub ..."
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Cited by 3 (2 self)
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to the subalgebra C0(T)AC0(T) of A, so that his general theory of proper actions gives a Morita equivalence between A ⋊α,r G and a generalised fixedpoint algebra A α. Here we investigate the functor (A, α) ↦ → A α and the naturality of Rieffel’s Morita equivalence, focusing in particular on the relationship
Proper actions, fixedpoint algebras and naturality in nonabelian duality
 J. Funct. Anal
"... Abstract. Suppose a locally compact group G acts freely and properly on a locally compact Hausdorff space X, and let γ be the induced action on C0(X). We consider a category in which the objects are C ∗dynamical systems (A, G, α) for which there is an equivariant homomorphism of (C0(X), γ) into the ..."
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Cited by 15 (8 self)
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) into the multiplier algebra M(A). Rieffel has shown that such systems are proper and saturated, and hence have a generalized fixedpoint algebra A α which is Morita equivalent to A×α,r G. We show that the assignment (A, α) ↦ → A α is functorial, and that Rieffel’s Morita equivalence is natural in a suitable sense. We
Generalized fixedpoint algebras of certain actions on crossed products
 Pacific J. Math
, 1995
"... Abstract: Let G and H be two locally compact groups acting on a C*algebra A by commuting actions λ and σ. We construct an action on A ×λ G out of σ and a unitary 2cocycle u. For A commutative, and free and proper actions λ and σ, we show that if the roles of λ and σ are reversed, and u is replaced ..."
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Cited by 13 (3 self)
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is replaced by u ∗, then the corresponding generalized fixedpoint algebras, in the sense of Rieffel, are strongMorita equivalent. We apply this result to the computation of the Ktheory of quantum Heisenberg manifolds. Introduction. Given two commuting actions λ and σ of locally compact groups G and H
Universal and exotic generalizes fixedpoint algebras for weakly proper actions and duality
 arXiv:1304.5697v2, 2013. 52 PAUL BAUM, ERIK GUENTNER, AND RUFUS WILLETT
"... ar ..."
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