Questions tagged [hilbert-modules]

Hilbert C*-modules are mathematical objects which generalise the notion of a Hilbert space (which itself is a generalisation of Euclidean space), in that they endow a linear space with an "inner product" which takes values in a C*-algebra

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Why is $\mathcal H_B\oplus \mathcal H_B\simeq \mathcal H_B$? Question about Hilbert $C^*$-modules and Kasparov's $KK$-Theory.

I am reading THE OPERATOR K-FUNCTOR AND EXTENSIONS OF $C^*$-ALGEBRAS To cite this article: G G Kasparov 1981 Math. USSR Izv. 16 513 Definition 1.1. $G$ isa fixed compact group satisfying the second ...
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31 views

How is $\mathcal L(E)$ graded? Question about Hilbert $B$-module

I am reading G.G.KASPAROV's THE OPERATOR AT-FUNCTOR AND EXTENSIONS OF C*-ALGEBRAS Let $B$ be a C*-algebra and let $E$ be a $B$-right-module. Assume there is a $B$-valued inner product on $E$ which (...
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Need references to learn about ideals of Hilbert $C^*-$ modules

I'm interested in learning about ideals of Hilbert $C^*-$modules and their relationship with ideals of linking algebra. Can someone please tell me some standard references(books/papers) for the same? ...
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29 views

Strict continuity of $^*$-homomorphism induced by interior tensor product

The following has been bugging me for a while, so I guess it's about time I asked for help with it on the internet. Consider the following set up, taken from halfway down p42 in Lance's Hilbert C$^*$-...
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28 views

Spatial isomorphism between type I factors

I'm reading a paper from Christopher J. Fewster about split property, and I'm trying to catch one of his assertions. He asserts that every type I factor $M$ (a concrete von Neumann algebra over $\...
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57 views

Factor a rank one operator on a Hilbert C*-module.

Let $A \subset \mathcal{L}(\mathcal{H}_0)$ be a concrete $C^*$-algebra and $X$ be a right Hilbert A-module. For each $x,y \in X$ we have rank one operators \begin{align} \theta_{x,y}: X & \to X \\ ...
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43 views

Extending a pre-Hilbert module operator over a pre-$C^*$-algebra to a Hilbert module over the completion

Let $A_0$ be a pre-$C^*$-algebra, i.e. $A_0$ satisfies all the criteria of a $C^*$-algebra except that it need not be complete. Then we can view $A_0$ as an inner product module over itself, i.e. a $\...
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Does localisation of a faithful retraction induce an isomorphism between adjointables of Hilbert modules?

The title is quite a mouthful, so let me develop some context. All of this is from the book on Hilbert Modules by C. Lance. If $A$ is a $C^*$-algebra, $M(A)$ its multiplier algebra and $B$ a sub-...
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103 views

Adjointable operators of a certain Hilbert module

In Lances book on Hilbert modules he states that if I have a C*-algebra $A$ and a (right) Hilbert module $E$ over $A$ then I can make the $n^{th}$ direct sum of $E$, denoted $E^n$, into a (right) ...
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70 views

Operator norm on a Hilbert module

Let $𝑋$ be a (right) Hilbert module over a $C^*$-algebra $𝐴$. Does it always hold that the $C^*$-norm of $a \in 𝐴$ is equal to its operator norm, when we consider $a$ as a bounded operator on $X$, ...
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73 views

Characterization of weak convergence in Hilbert $C^*$-modules?

Assume $M$ is a Hilbert $C^*$-module and $(x_n)^{\infty }_{n=1}$ a bounded sequence in $M$. Are these equivalence? $\langle x_n,y\rangle \to 0 $, for all $y\in M$. $(x_n)$ is convergent to $0$ in ...
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Doubt regarding Serre-Swan theorem and Hilbert modules

The Serre-Swan states that given a vector bundle $V$ with base $X$ (compact and Hausdorff), we have that $\Gamma(V)$ is a finitely generated projective $C(X)$-module. Moreover, all finitely generated ...
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Spectral decomposition of hilbert modules

Let $A$ be a $C^*$-algebra, $E$ a Hilbert $A$-module and $T:E\rightarrow E$ a bounded self-adjoint regular operator. Is there something like a spectral theorem for Hilbert modules that gives a ...
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104 views

Condition for an invertible operator on a Hilbert $C^*$-module

If $H_1$ and $H_2$ are Hilbert spaces and $T:H_1\rightarrow H_2$ a bounded linear operator, then one can show that $T$ is invertible if and only if there exists a constant $\alpha > 0$ such that $T^...
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174 views

Hilbert-Modules and Cohens factorization

Let $B$ be a $C^*$-algebra and $E$ a right Hilbert-$B$-Module. If I take an approximate unit $(e_i)_i$ in $B$ then for all $x \in E$ we have $x\cdot e_i \rightarrow x$ in $E$ by a simple calculation. ...
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179 views

Projection Lemma for Hilbert C-star modules

The projection lemma (for example, Rudin's functional analysis book theorem 12.4) says that if $M$ is a closed vector subspace of a $\mathbb{C}$-Hilbert space $\mathcal{H}$ then $$\mathcal{H}=M\oplus ...
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Why is a certain series in a Hilbert module well defined?

Suppose $(w_{\lambda})_{\lambda\in I}$ and $(\nu_{\lambda})_{\lambda\in I}$ ‎are orthonormal bases of a Hilbert module $E$ over the C*_algebra $\mathcal{A}$, such that $w_{\lambda}=u(\nu_{\lambda})$ ...
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139 views

Is the algebra of adjointable operators on a Hilbert module prime?

In abstract algebra, a nonzero ring $R$ is a prime ring if for any two elements $a$ and $b$ of R, $arb = 0$ for all $r$ in $R$ implies that either $a = 0$ or $b = 0.$ Or for any two ideals $A$ and $B$...
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226 views

What is the connection between Hilbert modules and tangent bundles in this paper?

A paper by Cipriani and Sauvageot, available at http://dx.doi.org/10.1016/S0022-1236(03)00085-5 shows that for many Dirichlet forms on $C^*$-algebras there is a derivation $\delta$ from the domain ...