##
**Representations of *-algebras, locally compact groups, and Banach *-algebraic bundles. Vol. 2: Banach *-algebraic bundles, induced representations, and the generalized Mackey analysis.**
*(English)*
Zbl 0652.46051

This second volume (for the first one see the review above) starts with chapter VIII “Banach *-Algebraic Bundles”. The authors develop foundations of this theory. It starts with a short account on mulipliers in Banach and \(C^*\)-algebras. For a topological group G a Banach alebraic bundle over G is a Banach bundle \(B=(B,\pi)\) together with an associative binary operation \(\cdot\) on B such that \(B_ xB_ y\subset B_{xy}\), where \(B_ x\) is a fiber over x, for all \(x,y\in G\), the product \(\cdot\) is bilinear on \(B_ x\times B_ y\to B_{xy}\), moreover \(\| b\cdot c\| \leq \| b\| \| c\|\) \((b,c\in B)\) and the map \(\cdot\) is continuous on \(B\times B\to B\) (the authors remark that the last condition does not follow from the previous one). Such a bundle is said to be saturated if the linear span of \(B_ xB_ y\) is dense in \(B_{xy}\). A Banach *-algebraic bundle is a Banach algebraic bundle with a map \(a\to a^*\) of B onto itself satisfying some natural conditions. Some operations (as retraction, semidirect products, central extensions) permit to obtain new Banach *-algebraic bundles from the old ones. If G is locally compact with Haar measure \(\lambda\) and \({\mathcal B}=(B,\pi,\cdot)\) then on (suitably defined) cross-sectional space \({\mathcal L}_ 1(\lambda,{\mathcal B})\) it is possible to define multiplication making of it a Banach algebra and called bundle convolution. The authors study this algebra and some of its generalizations (partial cross-sectional bundles). In § 8 the authors begin the representation theory of Banach algebraic bundles, first in general and later (§ 9) in the case of *-algebraic bundles and *-representations. They show that not only unitary representations but also so-called projective representations of groups can be included under the category of *-representations of Banach *-algebraic bundles. In § 11 they study integrable locally convex representations. A locally convex representation T of a Banach algebraic bundle B over a locally compact group G with Haar measure \(\lambda\) is said to be integrable if for each cross-section f of \({\mathcal B}\) with compact support there is a continuous linear operator \(\tilde T_ f\) with \(\alpha(\tilde T_ f\xi)=\int_{G}\alpha (T_{f(x)}\xi)d\lambda x\) for each \(\xi\) in X(T) and each \(\alpha\) in \((X(T))^*\). In some cases \(\tilde T\) in the above formula can be extended to a representation of \(L_ 1(\lambda,{\mathcal B})\) and it is said then an integrated form of T. Sections 12 and 13 are devoted to the recovery of T from its integrated form. In the next section 14 it is used in proving that under certain conditions \({\mathcal B}\) has enough irreducible *-representations to distinguish its points. In sections 15 and 16 the authors extend some facts on Banach algebras to the bundle situation. First they show that a non-degenerate *-representation of a Banach *-algebraic bundle can be extended to such a representation of the multiplier bundle. They give also a description of *-representations of semi-direct product bundles. In § 16 they introduce \(C^*\)-algebraic bundles (playing for the Banach algebraic bundles a similar rôle to that of \(C^*\)-algebras in the Banach algebra theory) and show that similarly as in the Banach algebra case to any Banach *-algebraic bundle there corresponds its bundle \(C^*\)-completion. If \({\mathcal B}\) is a Banach *-algebraic bundle over G, the \(C^*\)-completion (in the Banach algebra sense) of the \({\mathcal L}_ 1\) cross-sectional algebra \({\mathcal L}_ 1(\lambda,{\mathcal B})\) is called the cross-sectional \(C^*\)-algebra of \({\mathcal B}\) and it is denoted by \(C^*({\mathcal B})\). In § 17 the authors define \(C^*\)- bundle structure over an l.c. group G for a \(C^*\)-algebra A as a pair (\({\mathcal B},F)\), where \({\mathcal B}\) is a \(C^*\)-algebraic bundle over G and F is a *-isomorphism of \(C^*({\mathcal B})\) onto A. This structure is said to be saturated if B is saturated. It is shown that *-representations for A and \({\mathcal B}\) are essentially the same. Let \({\mathcal B}=(B,\pi,\cdot,*)\) be a Banach *-algebraic bundle over an l.c. group G and let M be a locally compact G-space. A system of imprimitivity for \({\mathcal B}\) over M is a pair (T,P), where T is a non-degenerate *-representation of B and P is a regular X(T)-projection-valued Borel measure on M such that \(T_ bP(W)=P(\pi (b)W)T_ b\) for all b in B and all Borel subsets W of M. In sections 18 and 19 the authors show that *-representations of so called transformation bundle (derived from \({\mathcal B})\) are in one-to-one correspondence with the imprimitivity systems for \({\mathcal B}\) and these objects can be constructed one from another. The concluding sections are § 20 Functionals of Positive Type on Banach *-Algebraic Bundles and § 21 The Regional Topology of *-Representations of Banach *-Algebraic Bundles.

The material of Chapter IX “Compact Groups” is almost entirely classical: Peter Weyl theorem, properties of the space of all finite- dimensional irreducible representations of G and properties of these representations, properties of locally convex representations of G, induced representations and Frobenius reciprocity theorem, classification of irreducible representations of SU(2) and SO(3).

Chapter X “Abelian Groups and Commutative Banach *-Algebraic Bundles” starts with an exposition of harmonic analysis on l.c.a. groups (the dual group and Pontryagin duality theorem, theorems of Stone, Bochner and Plancherel). Further the authors investigate the structure of saturated commutative Banach *-algebraic bundles. They show that the bundle structure of such a bundle \({\mathcal B}\) causes the structure space \(\hat {\mathcal B}\) to become a locally compact principal \(\hat G-\)bundle over the structure space \(\hat A\) of its unit fiber algebra \(A=B_ e\) and that every locally compact principle G-bundle arises in this way from some such \({\mathcal B}\). The authors obtain a duality result (a generalization of Pontryagin theorem): for saturated commutative \(C^*\)-algebraic bundles the correspondence \({\mathcal B}\to \hat {\mathcal B}\) is one-to-one and these bundles over G are essentially the same category of objects as locally compact principal G-bundles. The authors ask whether a similar result can be obtained in the non-commutative case.

The chapter XI “Induced Representations and the Imprimitivity Theorem” starts with the study of a concept of an operator inner product. This is a map \(V:(s,t)\to V_{s,t}\) from \(L\times L\) (L a linear space) to continuous linear operators of a Hilbert space X, which is linear in s, conjugate-linear in t and completely positive, i.e. \(\sum_{i,j\leq n}(V_{t_ i,t_ j}\xi_ i,\xi_ j)\geq 0\) for all \(n,t_ i\in L\), \(\xi_ i\in X\). Such a V is called non-degenerate if \(V_{s,t}\xi =0\) for all s and t implies \(\xi =0\). From a given operator inner product V one can obtain another such a product W called a deduced inner product from V. It is defined on the complex conjugate space \(\bar L\) of L. It turns out that repeating this procedure for W one obtains \(\tilde V\) on \(\bar{\bar L}=L\) and it is \(\tilde V\cong V\). This gives a duality theorem for operator inner products and it is the crux of abstract imprimitivity theorem. If A is a *-algebra and L is a left A-module, then L is said a Hermitian A-module (with respect to V) if \(V_{as,t}=V_{s,a^*t}\) where \(s,t\in L\), \(a\in A\). If A is a Banach *-algebra, then (under some assumptions) the action of A on L gives rise to a *-representation of A called the *-representation deduced from A-module L and V. The authors provide other formalisms for producing *-representations of *-algebras, called abstractly induced representations, in particular so called Rieffel inducing process by which one passes from a *-representation of one *-algebra to such a representation of an another *-algebra. The Frobenius inducing process for finite groups is a special case of the Rieffel inducing process. The process of passing from *-representation of one *-algebra to that of another one can be in certain cases reversed so that these two inducing operations are inverse to each other and provide an isomorphism between *-representation theories of the two algebras in question. This is the so called abstract imprimitivity theorem (it has also a more rich topological version). The inducing construction is applied to (suitably defined) \({\mathcal B}\)-positivity of *-representations. It is shown that if \({\mathcal B}\) is a saturated \(C^*\)-algebraic bundle over the locally compact group G and H is a closed subgroup of G then every *-representation of \({\mathcal B}_ H\) is positive with respect to \({\mathcal B}\). Also if \({\mathcal B},G,H\) are as above then for any *- representation S of \({\mathcal B}_ H\) the following conditions are equivalent: (i) S is \({\mathcal B}\)-positive; (ii) \(S| {\mathcal B}_ e\) is \({\mathcal B}\)-positive (Theorems 11.10 and 11.11; here \({\mathcal B}_ H\) is the reduction of \({\mathcal B}\) to H. The authors also ask whether these results are true without the saturation hypothesis). Further sections in this chapter are § 12 Elementary Properties of Induced Representations of Banach *-Algebraic Bundles, § 13 Restriction and Tensor Products of Induced Representations of Banach *-Algebraic Bundles, § 14 The Imprimitivity Theorem for Banach *-Algebraic Bundles (this is the chief result of this chapter. Let \({\mathcal B}\) be a Banach *-algebraic bundle over an l.c. group G and let H be a closed subgroup of G. The result says, roughly speaking, that any B-positive non-degenerate *-representation of \({\mathcal B}_ H\) induces a non-degenerate system of imprimitivity for \({\mathcal B}\) over G/H and conversely, any such a system is unitarily equivalent to a system induced by some non-degenerate \({\mathcal B}\)-positive *-representation of \({\mathcal B}_ H)\), § 15 A Generalized Mackey-Stone-von-Neumann Theorem, § 16 Conjugation of Representations, § 17 Non-involutory Induced Representations.

The last chapter XII is entitled “The Generalized Mackey Analysis”. The authors wirte: “This final chapter... is the climax toward which the earlier chapters have been pointing. Indeed, our choice of the material to be included in earlier chapters has been determined to a considerable extent by the requirements of the present chapter. Our interest in Banach *-algebraic bundles, especially saturated ones, is largely due to the fact that they seem to form the most general natural setting for the Mackey analysis... As far as the authors know, the present chapter is the first place in which the Mackey normal subgroup analysis for saturated bundles has appeared in print.” The authors start with description of the topic of this chapter in the case of a finite group G. The procedure for classifying the structure space of G in terms of a normal subgroup N of G is divided into three steps. A similar procedure is performed further in the more general context of saturated Banach *-algebraic bundles over locally compact groups. In § 6 it is given a complete description of the structure of an arbitrary saturated \(C^*\)-algebraic bundle (over an l.c. group) whose unit fiber \(C^*\)-algebra is of compact type. Such bundles are \(C^*\)-direct sums of certain specific \(C^*\)-algebraic bundles (theorem 6.19, it is due to J. M. G. Fell and it is its first publication). The chapter is closed with a section on saturated bundles over compact groups and with two sections devoted to various examples illustrating the Mackey analysis.

The bibliography contains almost 1300 positions.

The material of Chapter IX “Compact Groups” is almost entirely classical: Peter Weyl theorem, properties of the space of all finite- dimensional irreducible representations of G and properties of these representations, properties of locally convex representations of G, induced representations and Frobenius reciprocity theorem, classification of irreducible representations of SU(2) and SO(3).

Chapter X “Abelian Groups and Commutative Banach *-Algebraic Bundles” starts with an exposition of harmonic analysis on l.c.a. groups (the dual group and Pontryagin duality theorem, theorems of Stone, Bochner and Plancherel). Further the authors investigate the structure of saturated commutative Banach *-algebraic bundles. They show that the bundle structure of such a bundle \({\mathcal B}\) causes the structure space \(\hat {\mathcal B}\) to become a locally compact principal \(\hat G-\)bundle over the structure space \(\hat A\) of its unit fiber algebra \(A=B_ e\) and that every locally compact principle G-bundle arises in this way from some such \({\mathcal B}\). The authors obtain a duality result (a generalization of Pontryagin theorem): for saturated commutative \(C^*\)-algebraic bundles the correspondence \({\mathcal B}\to \hat {\mathcal B}\) is one-to-one and these bundles over G are essentially the same category of objects as locally compact principal G-bundles. The authors ask whether a similar result can be obtained in the non-commutative case.

The chapter XI “Induced Representations and the Imprimitivity Theorem” starts with the study of a concept of an operator inner product. This is a map \(V:(s,t)\to V_{s,t}\) from \(L\times L\) (L a linear space) to continuous linear operators of a Hilbert space X, which is linear in s, conjugate-linear in t and completely positive, i.e. \(\sum_{i,j\leq n}(V_{t_ i,t_ j}\xi_ i,\xi_ j)\geq 0\) for all \(n,t_ i\in L\), \(\xi_ i\in X\). Such a V is called non-degenerate if \(V_{s,t}\xi =0\) for all s and t implies \(\xi =0\). From a given operator inner product V one can obtain another such a product W called a deduced inner product from V. It is defined on the complex conjugate space \(\bar L\) of L. It turns out that repeating this procedure for W one obtains \(\tilde V\) on \(\bar{\bar L}=L\) and it is \(\tilde V\cong V\). This gives a duality theorem for operator inner products and it is the crux of abstract imprimitivity theorem. If A is a *-algebra and L is a left A-module, then L is said a Hermitian A-module (with respect to V) if \(V_{as,t}=V_{s,a^*t}\) where \(s,t\in L\), \(a\in A\). If A is a Banach *-algebra, then (under some assumptions) the action of A on L gives rise to a *-representation of A called the *-representation deduced from A-module L and V. The authors provide other formalisms for producing *-representations of *-algebras, called abstractly induced representations, in particular so called Rieffel inducing process by which one passes from a *-representation of one *-algebra to such a representation of an another *-algebra. The Frobenius inducing process for finite groups is a special case of the Rieffel inducing process. The process of passing from *-representation of one *-algebra to that of another one can be in certain cases reversed so that these two inducing operations are inverse to each other and provide an isomorphism between *-representation theories of the two algebras in question. This is the so called abstract imprimitivity theorem (it has also a more rich topological version). The inducing construction is applied to (suitably defined) \({\mathcal B}\)-positivity of *-representations. It is shown that if \({\mathcal B}\) is a saturated \(C^*\)-algebraic bundle over the locally compact group G and H is a closed subgroup of G then every *-representation of \({\mathcal B}_ H\) is positive with respect to \({\mathcal B}\). Also if \({\mathcal B},G,H\) are as above then for any *- representation S of \({\mathcal B}_ H\) the following conditions are equivalent: (i) S is \({\mathcal B}\)-positive; (ii) \(S| {\mathcal B}_ e\) is \({\mathcal B}\)-positive (Theorems 11.10 and 11.11; here \({\mathcal B}_ H\) is the reduction of \({\mathcal B}\) to H. The authors also ask whether these results are true without the saturation hypothesis). Further sections in this chapter are § 12 Elementary Properties of Induced Representations of Banach *-Algebraic Bundles, § 13 Restriction and Tensor Products of Induced Representations of Banach *-Algebraic Bundles, § 14 The Imprimitivity Theorem for Banach *-Algebraic Bundles (this is the chief result of this chapter. Let \({\mathcal B}\) be a Banach *-algebraic bundle over an l.c. group G and let H be a closed subgroup of G. The result says, roughly speaking, that any B-positive non-degenerate *-representation of \({\mathcal B}_ H\) induces a non-degenerate system of imprimitivity for \({\mathcal B}\) over G/H and conversely, any such a system is unitarily equivalent to a system induced by some non-degenerate \({\mathcal B}\)-positive *-representation of \({\mathcal B}_ H)\), § 15 A Generalized Mackey-Stone-von-Neumann Theorem, § 16 Conjugation of Representations, § 17 Non-involutory Induced Representations.

The last chapter XII is entitled “The Generalized Mackey Analysis”. The authors wirte: “This final chapter... is the climax toward which the earlier chapters have been pointing. Indeed, our choice of the material to be included in earlier chapters has been determined to a considerable extent by the requirements of the present chapter. Our interest in Banach *-algebraic bundles, especially saturated ones, is largely due to the fact that they seem to form the most general natural setting for the Mackey analysis... As far as the authors know, the present chapter is the first place in which the Mackey normal subgroup analysis for saturated bundles has appeared in print.” The authors start with description of the topic of this chapter in the case of a finite group G. The procedure for classifying the structure space of G in terms of a normal subgroup N of G is divided into three steps. A similar procedure is performed further in the more general context of saturated Banach *-algebraic bundles over locally compact groups. In § 6 it is given a complete description of the structure of an arbitrary saturated \(C^*\)-algebraic bundle (over an l.c. group) whose unit fiber \(C^*\)-algebra is of compact type. Such bundles are \(C^*\)-direct sums of certain specific \(C^*\)-algebraic bundles (theorem 6.19, it is due to J. M. G. Fell and it is its first publication). The chapter is closed with a section on saturated bundles over compact groups and with two sections devoted to various examples illustrating the Mackey analysis.

The bibliography contains almost 1300 positions.

Reviewer: W.Żelazko

### MSC:

46K10 | Representations of topological algebras with involution |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

46H15 | Representations of topological algebras |

22Dxx | Locally compact groups and their algebras |

43A65 | Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis) |

43-02 | Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis |

46M20 | Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) |

55R25 | Sphere bundles and vector bundles in algebraic topology |

22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |