Questions tagged [hilbert-modules]
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10
questions
3
votes
1
answer
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Property that follows from conditions involving slice maps on Hilbert module
Let $A,B$ be $C^*$-algebras and $E$ be a right $A$-Hilbert $C^*$-module. We can form the Hilbert $A\otimes B$ (minimal tensor product) module $E \otimes B$. If $\omega \in B^*$, there is a unique ...
6
votes
2
answers
442
views
Linear map between projective finitely generated Hilbert modules is adjointable
Let $A$ be a (unital) $C^*$-algebra and $X,Y$ right Hilbert $A$-modules which are finitely generated and projective. It seems to be well-known that if $T: X \to Y$ is an $A$-linear map, then $T$ is ...
3
votes
1
answer
112
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Unitary in adjointable operators associated with equivariant Hilbert module
Consider the following fragment from the article "Tannaka–Krein duality for compact quantum
homogeneous spaces. I. General theory" by De Commer and Yamashita:
How exactly is $\mathcal{E}\...
6
votes
0
answers
103
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$C(X)$-Fredholm operators and Atiyah-Jänich theorem
Let $X$ be a compact Hausdorff topological space and consider the Hilbert space $\ell^2(\mathbb N)$. As shown here, any $T\in C(X,\ell^2(\mathbb N))$ induces a $C(X)$-Fredholm operator
$$
\begin{array}...
4
votes
1
answer
162
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Reference request: decomposability of $\mathbb{G}$-Hilbert modules
Let $\mathbb{G}$ be a compact quantum group, $B$ be a $C^*$-algebra together with a right action
$$\beta: B \to B\otimes C(\mathbb{G})$$ which is a non-degenerate $*$-homomorphism satisfying $(\beta \...
2
votes
1
answer
165
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External tensor product of Hilbert modules
I am reading Lance's book "Hilbert $C^*$-modules". In particular, I want to understand how to construct the (external) tensor product of Hilbert $C^*$-modules. Consider the following ...
3
votes
1
answer
386
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Strict topology on the multiplier algebra
Let $A$ be a $C^*$-algebra. Let $M(A)$ be its multiplier $C^*$-algebras. The strict topology on $M(A)$ is given by
$$x_\lambda \to x \iff \forall a\in A: (\|x_\lambda a-xa\| + \|ax_\lambda - ax\| \to ...
2
votes
2
answers
179
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Unconditional Convergence of Positive Terms in a $C*$-algebra
I am reading the paper Frames and Outer Frames for Hilbert $C^*$-modules by L.J. Arambasic and D. Bakic. They have mentioned in passing, the following:
"...Since in each $C^*$-algebra, a ...
6
votes
0
answers
132
views
Examples of a full Hilbert C(X)-bimodule E such that the crossed product $C(X) \rtimes_E \mathbb{Z}$ is simple?
Let $A = C(X)$ be a commutative $C^*$-algebra. An example of a full finitely generated Hilbert $A$-bimodule E such that the crossed product $C(X) \rtimes_E \mathbb{Z}$, as defined by Abadie, Eilers ...
7
votes
1
answer
322
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Why is the definition of von Neumann trace independent of the choice of the Hilbert space?
A Hilbert module defined in "L^2-invariants: theory and applications to geometry and K-theory", Springer-Verlag, 2002, by W. Lück:
A Hilbert $\mathcal N(G)$-module $V$ is a Hilbert space $V$ ...