David P. Blecher, Wend Werner
In this first part of a study of ordered operator spaces, we develop the basic theory of `ordered C*-bimodules'. A crucial role is played by `open central tripotents', a JB*-triple variant of Akemann's notion of open projection.
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We generalize the notion of `diagonal' from the class of CSL algebras to masa bimodules. We prove that a reflexive masa bimodule decomposes as a sum of two bimodules, the diagonal and a module generalizing the w*-closure of the Jacobson radical of a CSL algebra. The latter module turns out to be reflexive, a result which is new even for CSL algebras. We show that the projection onto the direct summand contained in the diagonal is contractive and preserves compactness and reduces rank of operators. Stronger results are obtained when the module is the reflexive hull of its rank-one subspace. Key…
For an operator bimodule $X$ over von Neumann algebras $A\subseteq\bh$ and $B\subseteq\bk$, the space of all completely bounded $A,B$-bimodule maps from $X$ into $\bkh$, is the bimodule dual of $X$. Basic duality theory is developed with a particular attention to the Haagerup tensor product over von Neumann algebras. To $X$ a normal operator bimodule $\nor{X}$ is associated so that completely bounded $A,B$-bimodule maps from $X$ into normal operator bimodules factorize uniquely through $\nor{X}$. A construction of $\nor{X}$ in terms of biduals of $X$, $A$ and $B$ is presented. Various operator…
The injective tensor product of normal representable bimodules over von Neumann algebras is shown to be normal. The usual Banach module projective tensor product of central representable bimodules over an Abelian C$^*$-algebra is shown to be representable. A normal version of the projective tensor product is introduced for central normal bimodules.
Banach Gelfand Triples with Kernel Theorems via Equivalence Bimodules
本文通过等价双模构造Banach Gelfand三元组，证明有限生成投影Hilbert C*-模的核定理，并应用于Heisenberg模，得到时频分析的新核定理。
Banach bimodule-valued positive maps: Inequalities and induced representations
本文研究了Banach双模值正映射的诱导表示，并推导了相关不等式。
The duality principle for Gabor frames is one of the pillars of Gabor analysis. We establish a far-reaching generalization to Morita equivalent $C^*$-algebras where the equivalence bimodule is a finitely generated projective Hilbert $C^*$-module. These Hilbert $C^*$-modules are equipped with some extra structure and are called Gabor bimodules. We formulate a duality principle for standard module frames for Gabor bimodules which reduces to the well-known Gabor duality principle for twisted group $C^*$-algebras of a lattice in phase space. We lift all these results to the matrix algebra level an…
Recently Blecher and Kashyap have generalized the notion of W* modules over von Neumann algebras to the setting where the operator algebras are σ- weakly closed algebras of operators on a Hilbert space. They call these modules weak* rigged modules. We characterize the weak* rigged modules over nest algebras . We prove that Y is a right weak* rigged module over a nest algebra Alg(M) if and only if there exists a completely isometric normal representation ϕof Y and a nest algebra Alg(N) such that Alg(N)ϕ(Y)Alg(M) \subset ϕ(Y) while ϕ(Y) is implemented by a continuous nest homomorphism from M ont…
We give a short proof of the nuclearity property of a class of Cuntz-Pimsner algebras associated with a Hilbert A-bimodule M, where A is a separable and nuclear C*-algebra. We assume that the left A-action on the bimodule M is given in terms of compact module operators and that M is direct summand of the standard Hilbert module over A.
In this article, we construct a 2-shaded rigid ${\rm C}^*$ multitensor category with canonical unitary dual functor directly from a standard $λ$-lattice. We use the notions of traceless Markov towers and lattices to define the notion of module and bimodule over standard $λ$-lattice(s), and we explicitly construct the associated module category and bimodule category over the corresponding 2-shaded rigid ${\rm C}^*$ multitensor category. As an example, we compute the modules and bimodules for Temperley-Lieb-Jones standard $λ$-lattices in terms of traceless Markov towers and lattices. Translating…
For bi-Hilbertian $A$-bimodules, in the sense of Kajiwara--Pinzari--Watatani, we construct a Kasparov module representing the extension class defining the Cuntz--Pimsner algebra. The construction utilises a singular expectation which is defined using the $C^*$-module version of the Jones index for bi-Hilbertian bimodules. The Jones index data also determines a novel quasi-free dynamics and KMS states on these Cuntz--Pimsner algebras.
Suppose $A$ is a pro-C*-algebra. Let $L_{A}(E)$ be the pro-C*-algebra of adjointable operators on a Hilbert $A$-module $E$ and let $K_{A}(E)$ be the closed two sided $*$-ideal of all compact operators on $E$. We prove that if $E$ be a full Hilbert $A$-module, the innerness of derivations on $K_{A}(E)$ implies the innerness of derivations on $L_{A}(E)$. We show that if $A$ is a commutative pro-C*-algebra and $E$ is a Hilbert $A$-bimodule then every derivation on $K_{A}(E)$ is zero. Moreover, if $A$ is a commutative $σ$-C*-algebra and $E$ is a Hilbert $A$-bimodule then every derivation on $L_{A}…
We introduce Rokhlin properties for certain discrete group actions on $C^*$-correspondences as well as on Hilbert bimodules and analyze them. It turns out that the group actions on any $C^*$-correspondence $E$ with Rokhlin property induces group actions on the associated $C^*$-algebra $\mathcal O_E$ with Rokhlin property and the group actions on any Hilbert bimodule with Rokhlin property induces group actions on the linking algebra with Rokhlin property. Permanence properties of several notions such as nuclear dimension and $\mathcal D$-absorbing property with respect to crossed product of Hil…
We study the injective envelope I(X) of an operator space X, showing amongst other things that it is a self-dual C$^*-$module. We describe the diagonal corners of the injective envelope of the canonical operator system associated with X. We prove that if X is an operator $A-B$-bimodule, then A and B can be represented completely contractively as subalgebras of these corners. Thus, the operator algebras that can act on X are determined by these corners of I(X) and consequently bimodules actions on X extend naturally to actions on I(X). These results give another characterization of the multipli…
For two discrete metric spaces, $X$ and $Y$ we consider metrics on $X\sqcup Y$ compatible with the metrics on $X$ and $Y$. As morphisms from $X$ to $Y$ we consider the Roe bimodules, i.e. the norm closures of bounded finite propagation operators from $l^2(X)$ to $l^2(Y)$. We study the corresponding category $\mathcal M$, which is also a 2-category. We show that almost isometries determine morphisms in $\mathcal M$. We also consider the case $Y=X$, when there is a richer algebraic structure on the set of morphisms of $\mathcal M$: it is a partially ordered semigroup with the neutral element, wi…
We introduce a way of regarding Hilbert von Neumann modules as spaces of operators between Hilbert space, not unlike [Skei], but in an apparently much simpler manner and involving far less machinery. We verify that our definition is equivalent to that of [Skei], by verifying the `Riesz lemma' or what is called `self-duality' in [Skei]. An advantage with our approach is that we can totally side-step the need to go through $C^*$-modules and avoid the two stages of completion - first in norm, then in the strong operator topology - involved in the former approach. We establish the analogue of the …
Let $\mathcal{T}(\mathcal{N})$ be a nest algebra of operators on Hilbert space and let $\mathcal{L}$ be a weakly closed Lie $\mathcal{T}(\mathcal{N})$-module. We construct explicitly the largest possible weakly closed $\mathcal{T}(\mathcal{N})$-bimodule $\mathcal{J}(\mathcal{L})$ and a weakly closed $\mathcal{T}(\mathcal{N})$-bimodule $\mathcal{K}(\mathcal{L})$ such that \[ \mathcal{J}(\mathcal{L})\subseteq \mathcal{L} \subseteq \mathcal{K}(\mathcal{L}) +\mathcal{D}_{\mathcal{K}(\mathcal{L})}, \] $[\mathcal{K}(\mathcal{L}), \mathcal{T}(\mathcal{N})]\subseteq \mathcal{L}$ and $\mathcal{D}_{\mat…
We regard a right Hilbert C*-module X over a C*-algebra A endowed with an isometric *-homomorphism ϕ: A\to L_A(X) as an object X_A of the C*-category of right Hilbert A-modules. Following a construction by the first author and Roberts, we associate to it a C*-algebra O_{X_A} containing X as a ``Hilbert A-bimodule in O_{X_A}''. If X is full and finite projective O_{X_A} is the C*-algebra C*(X), the generalization of the Cuntz-Krieger algebras introduced by Pimsner. More generally, C*(X) is canonically embedded in O_{X_A} as the C*-subalgebra generated by X. Conversely, if X is full, O_{X_A} is …
Rigged modules over an operator algebra are a generalization of Hilbert modules over a $C^{\star}$-algebra. We characterize the rigged modules over an operator algebra $\mathcal A$ which are orthogonally complemented in $C_\infty(\mathcal A),$ the space of infinite columns with entries in $\mathcal A.$ We show that every such rigged module `restricts' to a bimodule of Morita equivalence between appropriate stably isomorphic operator algebras.
The paper makes the first steps into the study of extensions ("twisted sums") of noncommutative $L^p$-spaces regarded as Banach modules over the underlying von Neumann algebra $\mathcal M$. Our approach combines Kalton's description of extensions by centralizers (these are certain maps which are, in general, neither linear nor bounded) with a general principle, due to Rochberg and Weiss saying that whenever one finds a given Banach space $Y$ as an intermediate space in a (complex) interpolation scale, one automatically gets a self-extension $ 0\longrightarrow Y\longrightarrow X\longrightarrow …
We construct classes of von Neumann algebra modules by considering ``column sums" of noncommutative L^p spaces. Our abstract characterization is based on an L^{p/2}-valued inner product, thereby generalizing Hilbert C*-modules and representations on Hilbert space. While the (single) representation theory is similar to the L^2 case, the concept of L^p bimodule (p not 2) turns out to be nearly trivial.
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