#### DMCA

## Integrable and proper actions on C ∗ -algebras, and square-integrable representations of groups (1999)

Citations: | 15 - 0 self |

### Citations

600 |
Fundamentals of the Theory of Operator Algebras
- Kadison, Ringrose
- 1986
(Show Context)
Citation Context ...h the ultra-strong-∗ operator topology (p. 76 of [La]). Every bounded increasing net of self-adjoint elements in M(K) converges in the strong, so ultra-strong-∗ and strict, topologies (lemma 6.1.4 of =-=[KR1]-=-), and so for an action α of a locally compact group G, every integrable element is proper. That is, Pα = Mα with the notation of the previous sections. We will show that proper actions are closely re... |

335 |
Equivariant KK -theory and the Novikov conjecture,
- Kasparov
- 1988
(Show Context)
Citation Context ... C*-algebra is to have an equivariant morphism from a commutative C*algebra with proper action, whose image is central. This technique seems to have been first introduced by Kasparov, in section 3 of =-=[Ks]-=-. For more recent occurences see [GHT] and the references therein. Such a morphism is a special case of the situation of Theorem 5.3, so that we immediately obtain: 5.6 Corollary. Let α be an action o... |

230 | Classifying space for proper actions and K-theory of group C∗-algebras - Baum, Connes, et al. - 1993 |

136 |
C∗-Algebres et leurs Representations,
- Dixmier
- 1969
(Show Context)
Citation Context ...may be very small, even just {0}. But because τ is a trace, Mτ is a two-sided ideal in A, as is then Nτ. (See 6.2.1 of [Pe2].) By the semi-continuity of τ the image of Mτ in Hτ is dense (see 7.4.1 of =-=[D]-=-), and the representation of A on Hτ is non-degenerate. For a ∈ Mτ, b ∈ M + τ , and x ∈ G we have 〈Ux(a), b〉τ = τ(αx(a ∗ )b) = τ(b 1/2 αx(a ∗ )b 1/2 ) by proposition 5.2.2 of [Pe2]. But c ↦→ τ(b1/2cb1... |

104 |
Transforms Associated to Square Integrable Group Representations,
- Grossmann, Morlet, et al.
- 1985
(Show Context)
Citation Context ...t use of the theory of unbounded operators and their polar decomposition when dealing with this condition. A very similar argument, involving an irreducible representation, appears in the appendix of =-=[GMP]-=-. 7.15 Proposition. Let U be a unitary representation of G on H. Let ξ, η ∈ H, and suppose that cξη ∈ L 2 (G). Let Hξ denote the cyclic subspace generated by ξ, and replace η by its projection in this... |

104 |
Modular theory in operator algebras,
- Stratila
- 1981
(Show Context)
Citation Context ...should be meant by proper actions of quantum groups. Integrable actions of a fairly wide class of quantum groups (namely Kac algebras) acting on von Neumann algebras are discussed in section 18.19 of =-=[S]-=-. The action of any compact quantum group on a C ∗ -algebra should be proper, and indeed in this case one obtains the kind of relations between the fixed point algebra and the crossed product algebra ... |

87 |
On the regular representation of a nonunimodular locally compact group,
- Dufflo, Moore
- 1976
(Show Context)
Citation Context ...f an isometry from H. In particular, every element of H would be in the range of N, so that cξζ ∈ L 2 (G) for all ζ ∈ H. When this is combined with part iii of the restatement in [BT] of theorem 3 of =-=[DM]-=-, this says that if ξ is “admissible” [BT], then ξ ∈ BU. The next proposition ties the situation a bit more closely to the discussion of the previous sections. Its proof is an immediate application of... |

72 | Représentation des produits croisés d’algèbres de groupoïdes - Renault - 1987 |

65 |
Topologie Générale,
- Bourbaki
- 1961
(Show Context)
Citation Context ... of G on a locally compact space M, then α is said to be proper if the map (m, x) → (m, αx(m)) from M × G to M × M is proper, in the sense that preimages of compact sets are compact. It is well known =-=[Bo]-=- that in this case the orbit space, M/α, with the quotient topology from M, is locally compact Hausdorff. The functions in C∞(M/α) can be viewed as functions in Cb(M) which are α-invariant. Here Cb(M)... |

60 | Unbounded elements affiliated with C∗-algebras and non-compact quantum groups, - Woronowicz - 1991 |

57 |
C*-algebras of transformation groups with smooth orbit space,
- Green
- 1977
(Show Context)
Citation Context ...e orbits be closed and the stability subgroups be compact. This is seen by the following example, which is a slightly more complicated version of the example at the very end of Philip Green’s article =-=[G]-=-. We make our example yet slightly more complicated than needed here so that we can also use it in the next section to illustrate a point there. 1.18 Example. The space M is a closed subset of R 3 , a... |

55 | Duality theory for covariant systems - Landstad - 1979 |

51 |
Coherent states and their generalizations: A mathematical overview
- Ali, Antoine, et al.
- 1995
(Show Context)
Citation Context ... The second equality says exactly that Hξ is a “reproducing-kernel Hilbert space” on G, with reproducing kernel cξξ. The third equality says that the map x ↦→ Lxcξξ is a “coherent state” for Hξ. (See =-=[Al]-=- for a recent review of coherent states, with many interesting examples.) Suppose now that ζ, ω ∈ BU. Then from 8.2 but with the roles of the vectors interchanged, and from the definition of γ(ζ, ω), ... |

49 |
A survey of foliations and operator algebras”, Operator algebras and applications
- Connes
- 1982
(Show Context)
Citation Context ...losely related to the definition of bounded elements in Hilbert algebras [D,Rf1,Pj,Cm]. Compare also with Connes’ treatment of square-integrable representations of foliations beginning on page 573 of =-=[Cn]-=-. (For a recent variation see definition 1.3 of [Bi].) It is clear that BU is a linear subspace, possibly not closed, in H. If ξ ∈ BU and x ∈ G, then for η ∈ H cUxξ,η(y) = 〈UyUxξ, η〉 = cξη(yx). Thus (... |

48 |
Wavelets from square-integrable representations
- Bernier, Taylor
- 1996
(Show Context)
Citation Context ... must be a multiple of an isometry from H. In particular, every element of H would be in the range of N, so that cξζ ∈ L 2 (G) for all ζ ∈ H. When this is combined with part iii of the restatement in =-=[BT]-=- of theorem 3 of [DM], this says that if ξ is “admissible” [BT], then ξ ∈ BU. The next proposition ties the situation a bit more closely to the discussion of the previous sections. Its proof is an imm... |

45 |
The flow of weights on factors of type
- Connes, Takesaki
- 1977
(Show Context)
Citation Context ...ly different point of view. 1. Integrable Actions. The material discussed here is very close to material on integrable actions in the von Neumann algebra literature. See for example definition 2.1 of =-=[CT]-=-, the introduction to [Ld], [Pa], and 18.20 of [S]. Here we stress the C∗-algebra version of integrable actions, so that we can contrast it with the notion of proper actions which we discuss in the ne... |

36 |
higher relative commutants and the fusion algebra associated to a subfactor, Operator algebras and their applications
- Bisch, Bimodules
- 1994
(Show Context)
Citation Context ... in Hilbert algebras [D,Rf1,Pj,Cm]. Compare also with Connes’ treatment of square-integrable representations of foliations beginning on page 573 of [Cn]. (For a recent variation see definition 1.3 of =-=[Bi]-=-.) It is clear that BU is a linear subspace, possibly not closed, in H. If ξ ∈ BU and x ∈ G, then for η ∈ H cUxξ,η(y) = 〈UyUxξ, η〉 = cξη(yx). Thus (7.7) ‖cUxξ,η‖2 = ∆(x) −1/2 ‖cξη‖2. Consequently BU i... |

34 | Induced representations of crossed products by coactions - Mansfield - 1991 |

30 |
Applications of the Connes Spectrum to C∗-dynamical systems
- Olesen, Pedersen
- 1982
(Show Context)
Citation Context ...b)} converges to m strictly. Thus b is P-proper as desired. □ The following lemma is motivated by, and very closely related to, proposition 6.6 of [E1]and to the comments in 7.8.4 of [Pe2] and 2.4 of =-=[OP1]-=-. See also lemma 3.5 of [Qg] and lemma 4.1 of [Ku]. 13 □2.7 Key Lemma. Let P = {pλ} be an increasing net of positive operators from B to M(A). Then the cone P + P of P-proper elements of B+ is heredi... |

18 |
Equivariant K-theory for proper actions
- Phillips
- 1988
(Show Context)
Citation Context ... C∞(M/α) can be viewed as functions in Cb(M) which are α-invariant. Here Cb(M) is the algebra of bounded continuous functions on M, and it is the multiplier algebra of C∞(M). It is well-known (2.4 of =-=[Pn]-=-) that if h ∈ Cc(M), and if we set ∫ ψ(h)(m) = h(α −1 x (m))dx for every m ∈ M, then ψ(h) is a function in C∞(M/α) ⊆ Cb(M). It is natural to write ∫ ψ(h) = αx(h)dx, but as before, the integrand is not... |

16 |
Square-integrable representations of non-unimodular groups,
- Carey
- 1976
(Show Context)
Citation Context ...le of the identity operator. We denote the scalar by γ(ξ, η), so that TξT ∗ η = γ(ξ, η)IH. We wish to obtain a more revealing expression for γ. We follow the general outline of the treatment given in =-=[Ca]-=-, but our details are more elementary because of our use of BU. By suitably normalizing ξ, we can arrange that T ∗ ξ is an isometry. Then T ∗ ξ Tξ is a projection operator on L 2 (G), intertwining L. ... |

13 | Generalized fixed-point algebras of certain actions on crossed-products - Abadie |

12 |
Landstad duality for C∗-coactions
- Quigg
- 1992
(Show Context)
Citation Context ...Thus b is P-proper as desired. □ The following lemma is motivated by, and very closely related to, proposition 6.6 of [E1]and to the comments in 7.8.4 of [Pe2] and 2.4 of [OP1]. See also lemma 3.5 of =-=[Qg]-=- and lemma 4.1 of [Ku]. 13 □2.7 Key Lemma. Let P = {pλ} be an increasing net of positive operators from B to M(A). Then the cone P + P of P-proper elements of B+ is hereditary. Proof. Let b ∈ P + P ,... |

11 |
On conditional expectations arising from group actions
- Frank, Manuilov, et al.
(Show Context)
Citation Context ...grable. But if we identify M/α with the fundamental domain [0, 4), then it is easily seen that ∑ x∈G αx(f) is χ (0,2), which is not in M(A). We remark that a related example appears as example 2.4 of =-=[FMT]-=-. We now have the corresponding version of Theorem 3.4. It relates our situation to definition 3.4 of [Qg] and section 1 of [QR]. 4.3 Theorem. Let n ∈ (M(A)e)+. Then n ∈ ˜ P + α if and only if there i... |

11 |
Algebras and Their Automorphism Groups
- Dixmier, C-Algebras
- 1977
(Show Context)
Citation Context ... strong operator topology) if they are in a von Neumann algebra (5.1.4 of [KR2]). Thus if we view pλ as taking values in the double-dual (or “enveloping”) von Neumann algebra, A ′′ , of A (see 3.7 of =-=[Pe2]-=-), we will have such convergence. Let us examine this situation a bit more generally. 1.1 Definition. Let N be some von Neumann algebra, and let P = {pλ} be an increasing net of positive maps from a C... |

10 | P.: An introduction to ergodic theory - Walter - 1982 |

10 | Unconditional integrability for dual actions
- Exel
- 1999
(Show Context)
Citation Context ...the classical notion of proper actions on spaces is one of continuing strong interest [BCH, GHT]. The definition of proper actions which we give here was strongly stimulated by a recent paper of Exel =-=[E]-=- (which in turn built on [Rf7]). In fact our definition almost appears explicitly in [E]. The main difference is that here we emphasize the order properties of C ∗ -algebras while in [E] the emphasis ... |

9 |
Induced C-algebras and Landstad duality for twisted coactions
- Quigg, Raeburn
- 1995
(Show Context)
Citation Context ... that it may be useful in our more general case of X to consider eventually various subalgebras of B. For example, our considerations can be applied to the induced C*-algebras studied for instance in =-=[QR]-=-. Here one has both an action α on A and an action, τ, on a space 3M, and one considers the subalgebra of Cb(M, A) consisting of the functions f such that f(τ −1 x (m)) = αx(f(m)). For any positive λ... |

8 |
Multiplicateurs non-bornés, Thése 3éme Cycle
- Baaj
- 1980
(Show Context)
Citation Context ...nd let τ be a trace on A which we now suppose to be actually α-invariant. Suppose that d is an unbounded positive invertible operator affiliated with A in the sense of Woronowicz [Wo]. (See also Baaj =-=[Ba]-=-.) For our purposes this means that we have a morphism, say θ, from D = C∞(R) to A (that is, a ∗-homomorphism from D into M(A) such that θ(D)A is dense in A) together with a strictly positive δ ∈ C(R)... |

8 |
of the theory of operator algebras
- Fundamentals
- 1997
(Show Context)
Citation Context ...ions f such that f(τ −1 x (m)) = αx(f(m)). For any positive λ ∈ L1 (X) define a linear map, pλ, from B to A by ∫ pλ(f) = f(x)λ(x)dx. It is easily seen that pλ is positive, in fact completely positive =-=[KR2]-=-, and of norm ‖λ‖1. We would like to have the flexibility of having λ range over characteristic functions of compact sets, or over continuous functions which approximate them. It is thus convenient fo... |

8 |
A construction for weights on C∗-algebras. Dual weights on C∗-crossed products.
- Quaegebeur, Verding
- 1999
(Show Context)
Citation Context ...Uµa = Uxa dµ(x) , in terms of the C ∗ -norm of A. Then Uµa ∈ Pa, and ∫ ψα(Uµa) = ( G ∆(y) −1 2 dµ(y))ψ(a) . 27sdifficult even in the case of ordinary weights (in contrast to traces). See lemma 3.1 of =-=[QV]-=- for a fairly restrictive hypotheses, “regularity” (also discussed in [Ku]), under which one can prove this strong continuity for weights. It is natural to expect that U is strongly continuous on vect... |

7 |
actions of groups on C∗-algebras, Mappings of operator algebras
- Proper
- 1990
(Show Context)
Citation Context ...y 22D10, 46L55; Secondary 54H20. The research was supported in part by National Science Foundation grant DMS96-13833. 1 Typeset by AMS-TEXI had earlier given a tentative definition of proper actions =-=[Rf7]-=-, which was adequate to treat some interesting examples. But that definition assumed the existence of a dense subalgebra with certain properties, and so was not intrinsic. The definition proposed here... |

5 | Regular C∗-valued weights
- Kustermans
(Show Context)
Citation Context ...arlier notions of integrable elements discussed in [Ld, OP1, OP2, Qg, QR]. It is also closely related to the notion of C ∗ -valued weights on C ∗ -algebras which was introduced recently by Kustermans =-=[Ku]-=-, for fairly different reasons involving Haar measures for quantum groups. (I thank Kustermans for some helpful comments on this matter.) One can in turn ask what should be meant by proper actions of ... |

4 |
of the Connes spectrum to C∗-dynamical systems
- Applications
- 1980
(Show Context)
Citation Context ...terms of linear functionals on A. It is related to the definition of ˆα-integrable elements given on page 269 of [Pe2], which originated in [OP1, OP2]. But we call attention to the note at the end of =-=[OP2]-=- which points out that the definition in [Pe2] is too strong, since it should only consider the dual, B ′ , of B (in the notation of [Pe2]), not of M(B). For some later variants see [QR] and its refer... |

4 | A note on square-integrable representations - Phillips - 1975 |

4 | equivalence for operator algebras, Operator algebras and applications, Part I - Morita - 1980 |

4 |
equivalences between fixed point algebras and crossed products
- Ng, Morita
- 1999
(Show Context)
Citation Context ...y compact quantum group on a C ∗ -algebra should be proper, and indeed in this case one obtains the kind of relations between the fixed point algebra and the crossed product algebra which one expects =-=[Ng]-=-. One can also ask about proper actions of groupoids on C ∗ -algebras, extending the notion of proper actions of groupoids on spaces given in [Re]. In section 1 we deal with integrable actions, while ... |

3 |
equivalence and spectral theory for integrable automorphism groups of C∗-algebras
- Morita-Rieffel
(Show Context)
Citation Context ... will be strongly Morita equivalent to the entire reduced crossed product algebra exactly if the action on X is free.) The outcome of our discussion will be far from satisfactory. In particular, Exel =-=[E2]-=- has shown that the candidate for “generalized fixed-point algebra” which I had suggested in the first version of this paper is often too big. (See our discussion following Theorem 8.5.) In fact, Exel... |

3 |
representations of group extensions: an algebraic approach to the theory of
- Unitary
- 1979
(Show Context)
Citation Context ... thus a countable discrete set, and so it cannot be strongly Morita equivalent to a unital C ∗ -algebra, since strongly Morita C ∗ -algebras have homeomorphic primitive ideal spaces (corollary 3.3 of =-=[Rf3]-=-). But let g be as in Example 2.10. It is seen there that ψ(g) = 1. So the closure of the range of ψ is a unital C ∗ -algebra, and thus cannot be strongly Morita equivalent to A ×α G. Of course the di... |

3 |
Equivariant KK-theory and C∗-extensions, K-theory 19
- Thomsen
- 2000
(Show Context)
Citation Context ...nsequently the above expression can be rewritten as ∫ f1( a(f2αx(g1))αx(b)αx(g2)dx)h, which is manifestly in A0. □ It is not clear to me how Thomsen’s definition of a K-proper action, given in 9.1 of =-=[T]-=-, relates to our present considerations, though it has some relation to [GHT]. 6. Strong Morita equivalence. In this section we will discuss what one might take as the “generalized fixed-point algebra... |

2 |
Systèmes Hilbertiens à gauche et représentation de Gelfand-Segal, in Operator Algebras and Group representations vol 1
- Combes
(Show Context)
Citation Context ...tself by U. 7.8 Definition. We will say that a unitary representation U of G on H is square-integrable if BU is dense in H. This is exactly the special case for groups of Combes’ definition in 1.7 of =-=[Cm]-=- for left Hilbert algebras. We will see that this definition is equivalent to the more traditional definitions in those situations where they have been given. But conditions 7.3 and 7.4, which do not ... |

2 | Square-integrable representations of Hilbert algebras - Rieffel - 1969 |

2 |
of C∗-algebras deformed by actions of Rd
- K-groups
- 1993
(Show Context)
Citation Context ..., namely its restriction to the closure of Pα. We show now that all of the examples successfully treated by the definition of [Rf7] are examples of proper actions in the sense of Definition 4.5. (See =-=[Rf8]-=- and [M] for further such examples in addition to those already described in [Rf7].) This already gives a substantial supply of interesting examples. The main theorem of [E1] provides yet a further cl... |

2 | Square-integrable factor representations of locally compact groups - Rosenberg - 1978 |

1 |
Unconditional integrability for dual actions, math.OA/9504051
- Exel
- 1999
(Show Context)
Citation Context ...of proper actions on spaces is one of continuing strong interest [BCH, GHT].) Actually, the definition of proper actions which we give here was strongly stimulated by a slightly earlier paper of Exel =-=[E1]-=- (which in turn built on [Rf7]). In fact our definition almost appears explicitly in [E1]. The main difference is that here we emphasize the order properties of C ∗ -algebras while in [E1] the emphasi... |

1 |
Equivariant E-theory for C*-algebras, preprint
- Guentner, Higson, et al.
(Show Context)
Citation Context ... morphism from a commutative C*algebra with proper action, whose image is central. This technique seems to have been first introduced by Kasparov, in section 3 of [Ks]. For more recent occurences see =-=[GHT]-=- and the references therein. Such a morphism is a special case of the situation of Theorem 5.3, so that we immediately obtain: 5.6 Corollary. Let α be an action of G on a C*-algebra A. Let β be a prop... |

1 |
Square integrable primary representations
- Moore
- 1977
(Show Context)
Citation Context ... independent of the choice of ξ. But one can check that, as expected from [DM], K is ∆ −1 -semi-invariant, reflecting the situation for the left regular representation seen earlier. We remark that in =-=[Mo]-=- Moore has given orthogonality relations for factor squareintegrable representations. But his orthogonality relations are not for the coefficient functions as defined here. So it is not clear to me ho... |

1 |
Integrable group actions on von Neumann algebras
- Paschke
- 1977
(Show Context)
Citation Context ...ntegrable Actions. The material discussed here is very close to material on integrable actions in the von Neumann algebra literature. See for example definition 2.1 of [CT], the introduction to [Ld], =-=[Pa]-=-, and 18.20 of [S]. Here we stress the C∗-algebra version of integrable actions, so that we can contrast it with the notion of proper actions which we discuss in the next section. Since every proper a... |

1 |
Measure theory for C*-algebras, Math.Scand
- Pedersen
- 1966
(Show Context)
Citation Context ...onnection with all this is that if a ∈ M, it does not follow in general that |a| ∈ M (even if a = a ∗ ). This difficulty already occurs with ordinary weights. See the example following theorem 2.4 of =-=[Pe1]-=-. This makes it awkward to define an “L 1 -norm” on M using ϕ. In Theorem 8.9 we will give a simple explicit example in which this difficulty occurs exactly in our context, namely for an integrable ac... |

1 |
On the C*-algebras of Foliations of the Plane
- Wang
- 1987
(Show Context)
Citation Context ...ble. On the other hand, if the sequence {Ln} is bounded, then one can check that α is integrable. We remark that by examining the foliations of the plane (which come from actions of R), as studied in =-=[Wn]-=-, we obtain an abundance of examples of integrable actions α on C∞(M) such that M/α is not Hausdorff (but the actions are free, with closed orbits). Thus, integrability does not imply that M/α is Haus... |

1 | Applications of the Connes spectrum to C*-dynamical systems - unknown authors - 1996 |