论文标题
诱导，吸收和弱的 * - 代表 * - 代数捆绑
Induction, absorption and weak containment of *-representations of Banach *-algebraic bundles
论文作者
论文摘要
给定一个倒束$ \ nathcal {b} = \ {b_t \} _ {t \ in G} $上的lch组和一个封闭的子组$ h \ subset g，$，我们表明所有 * - 占$ \ \ \ \ \ \ \ \ \ natcal {b} b} _h：h： * - $ \ Mathcal {b} $的代表通过Fell的归纳过程；我们将其描述为通过a*-homorphism $ q^{\ Mathcal {b}} _ H \ colon C^*（\ Mathcal {B}）\ to \ Mathbb {B}（x___ { $ c^*_ h（\ Mathcal {b}）：= q^{\ Mathcal {b}} _ H（c^*（\ Mathcal {b}））$中间为$ c^*（\ Mathcal {b}） $ c^*_ {r}（\ Mathcal {b}）= C^*_ {\ {e \}}（\ Mathcal {b}）$，因为每个包含子组$ h \ subset k \ subset g $ subset g $ subset g $ subset g $给出了独特的映射$ q^{ c^*_ k（\ Mathcal {b}）\ to C^*_ H（\ Mathcal {B}）$，以至于$ q^{\ Mathcal {b}} _ {在$ \ Mathcal {b}，$ $ g，\ h $和$ k $（例如饱和，核能或弱遏制）上查找条件，暗示$ q^{\ Mathcal {b}} _ {hk} _ {hk} $是忠实的。我们的主要工具之一是FLEL的吸收原理（用于饱和束）的混合物，以及exel和ng的结果，用于减少横截面C* - 代数。我们还表明，鉴于$ g/h上的$ \ mathcal {b} $ for $ g/h，$ g/h，如果$ h $开放或在$ g中具有开放率的净量表，则$ t $ then $ t $弱属于$ \ mathcal not $ \ b} $（Iff，如果是$ g的，饱和）。给定$ g的正常和闭合子组，$ $ h \ subset k，$我们在$ g/k $上构建了一个fell束$ \ nathcal {c} $，以至于$ c^*_ r（\ nathcal {c}）= c^*_ h（\ mathcal {b}）。如果两个$ q^{\ mathcal {c}} _ {\ {e \}} $和$ q^{\ MATHCAL {b} _K} _H $是。
Given a Fell bundle $\mathcal{B}=\{B_t\}_{t\in G}$ over a LCH group and a closed subgroup $H\subset G,$ we show that all the *-representations of $\mathcal{B}_H:=\{B_t\}_{t\in H}$ can be induced to *-representations of $\mathcal{B}$ by means of Fell's induction process; which we describe as induction via a *-homomorphism $q^{\mathcal{B}}_H\colon C^*(\mathcal{B})\to \mathbb{B}(X_{C^*(\mathcal{B}_H)}).$ The quotients $C^*_H(\mathcal{B}):=q^{\mathcal{B}}_H(C^*(\mathcal{B}))$ are intermediate to $C^*(\mathcal{B})= C^*_G(\mathcal{B})$ and $C^*_{r}(\mathcal{B})=C^*_{\{e\}}(\mathcal{B})$ because every inclusion of subgroups $H\subset K\subset G$ gives a unique quotient map $q^{\mathcal{B}}_{HK}\colon C^*_K(\mathcal{B})\to C^*_H(\mathcal{B})$ such that $q^{\mathcal{B}}_{HK}\circ q^{\mathcal{B}}_K=q^{\mathcal{B}}_H.$ All along the article we try to find conditions on $\mathcal{B},$ $G,\ H$ and $K$ (e.g. saturation, nuclearity or weak containment) that imply $q^{\mathcal{B}}_{HK}$ is faithful. One of our main tools is a blend of Fell's absorption principle (for saturated bundles) and a result of Exel and Ng for reduced cross sectional C*-algebras. We also show that given an imprimitivity system $\langle T,P\rangle$ for $\mathcal{B}$ over $G/H,$ if $H$ is open or has open normalizer in $G,$ then $T$ is weakly contained in a *-representation induced from $\mathcal{B}_H$ (even if $\mathcal{B}$ is not saturated). Given normal and closed subgroups of $G,$ $H\subset K,$ we construct a Fell bundle $\mathcal{C}$ over $G/K$ such that $C^*_r(\mathcal{C})=C^*_H(\mathcal{B}).$ We show that $q^{\mathcal{B}}_H$ is faithful if and only if both $q^{\mathcal{C}}_{\{e\}}$ and $q^{\mathcal{B}_K}_H$ are.
