Questions tagged [hilbert-modules]

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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 ...
Andromeda's user avatar
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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 ...
Andromeda's user avatar
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3 votes
1 answer
112 views

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}\...
Andromeda's user avatar
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6 votes
0 answers
103 views

$C(X)$-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}...
Mezzovilla's user avatar
4 votes
1 answer
162 views

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 \...
J. De Ro's user avatar
  • 508
2 votes
1 answer
165 views

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 ...
Andromeda's user avatar
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3 votes
1 answer
386 views

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 ...
Andromeda's user avatar
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2 votes
2 answers
179 views

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 ...
Kurome's user avatar
  • 155
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 ...
Marie Anderlecht's user avatar
7 votes
1 answer
322 views

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 $V$ ...
Meisam Soleimani Malekan's user avatar