Tagged Questions

The subject of operator algebras is primarily C*-algebras and von Neumann algebras, and associated topics. It also includes more general algebras of operators on Hilbert space, and may include algebras of operators on other topological vector spaces. It is related to but distinct from the subjects ...

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K-theory for non-separable C*-algebras

Let $\kappa$ be an uncountable cardinal. What is the K-theory for the C*-algebras $\mathcal{K}(\ell_2(\kappa))$ and $\mathcal{B}(\ell_2(\kappa))$, of, respectively, compact and bounded operators on ...
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Combinatorial total space for finitely generated torsion-free groups?

Motivation: I'm an operator algebraist and I'm looking for an answer to the main question in order to build non-trivial spectral triples for a class (as large as possible) of discrete groups. $\to$ ...
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Products in $C^*$-algebra $K$-theory

Let $A_1$ and $A_2$ be unital $C^*$-algebras. If $p_1 \in M_{n_1}(A_1)$ and $p_2 \in M_{n_2}(A_2)$ are projections then $p_1 \otimes p_2 \in M_{n_1 n_2}(A_1 \otimes A_2)$ is also a projection, ...
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Compactum of Banach algebra

I need an example of Banach algebra $A$ and a left non-trivial closed ideal $I$ with all of following properties: There exists a bounded approximate identity in $I$ for $I$ i.e., a net ...
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Masas in quotients

Let $A$ be a von Neumann algebra and let $B$ be a norm-closed ideal of $A$ (but not necessarily WOT-closed). What one has to assume about $A$ and $B$ to ensure that if $M\subset A$ is a maximal ...
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Direct limits of completely positive maps on $C^*$-algebras vs. operator systems

I believe I've heard, as part of the "lore," that the category (operator systems, completely positive maps) has direct limits, whereas the category ($C^*$-algebras, completely positive maps) does not. ...
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A nilpotent element of an algebra which does not lie in the span of commutator elements.

What is an example of a $C^{*}$ algebra such that the span of nilpotent elements is not a sub vector space of the span of commutator elements. Obviously any such $C^{*}$ algebra would be a ...
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Finite dimensional Banach algebras whose $K_{0}$ group is a non trivial finite group

Motivated by this question we ask Is there a finite dimensional Banach algebra $A$ such that $K_{0}(A)$ is a nontrivial finite group? I understand from the above link and this post that any ...
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$\mathcal{H}_B$ is isomorphic to $B$ as Hilbert $B$-modules if and only if $B$ is a stable C*-algebra

Let $B$ be a C*-algebra. I want to prove that $\mathcal{H}_B$ is isomorphic to $B$ as Hilbert $B$-modules if and only if $B$ is stable, that is, $\mathcal{K} \otimes B$ is isomorphic to $B$ as ...
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Equivalence of categories ($c^*$ algebras <-> topological spaces)

I try to use a littlebit category theory to have a better overview of the results in the theory of $c^*$-algebras, but I really have to read an introduction to category theory because I know almost ...
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Reduced $C^*$-algebra of a direct product of locally compact groups

Is it true that $$C^*_r(G_1\times G_2)=C^*_r(G_1)\otimes_{\min}C^*_r(G_2)$$ for locally compact groups $G_1$ and $G_2$? I have managed to prove that it holds for discrete groups (see below), but as ...
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Pureness of Vector States

How does one show that irreducibility is equivalent to a vector state being pure? In what follows I will fill in the details of the question: Let $\mathcal{H}$ denote a Hilbert space and let ...
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Calculating Norms for the Transpose

How does one show that the transpose mapping $T: \mathcal{M}_{n} \to \mathcal{M}_{n}$, given by $T(a)=a^{t}$, has norm $||T||=1$ but completely bounded norm $||T||_{CB}=n$? Notation. ...
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Examples of $C^*$-algebras in Noncommutative Geometry from A. Connes

Question I am working on $C^*$-algebras and I've been given Alain Connes's book Noncommutative Geometry. I am having troubles with understanding the examples on pages 91-93 (86-88 in the printed ...
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Creating Bratteli diagrams for Riesz groups

The Effros-Handelman-Shen-theorem tells you that Riesz groups are the same as dimension groups -- i.e. any ordered, unperforated abelian group with the Riesz interpolation property can be realised as ...
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Maximal abelian subalgebras of SAW*-algebras

Pedersen distilled the following class of C*-algebras which he termed SAW*-algebras: A C*-algebra $A$ is an SAW*-algebra if for each pair of orthogonal, positive elements $x,y\in A$, there exists a ...
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Topologizing $\mathcal{L}(\mathcal{H}) / \mathcal{S}^p(\mathcal{H})$

Given a separable Hilbert space $\mathcal{H}$ I would like to know how one could topologize the quotient algebra $\mathcal{L}(\mathcal{H}) / \mathcal{S}^p(\mathcal{H})$? Here ...
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Short exact sequence involving mapping cone, cone, suspension of $C^*$-algebras

This is part of exercise 6.N in Wegge-Olsen's book '$K$-theory and $C^*$-algebras'. In the following, $A$ and $B$ are $C^*$-algebras, $\alpha:A\rightarrow B$ is a surjective $C^*$ morphism with kernel ...
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A question on the completely positive maps and manifold structure

I was reading a paper in which the curvature and Euler characteristic of a completely positive map (in finite dimensions). Let $$\Phi(X)=\sum_{j=1}^nV_jXV_j^*$$ be a ...
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Motivations for and connections between the topologies of Vietoris, Fell and Chabauty

My main interest is in the Chabauty topology on the space of closed subgroups of a locally compact topological group, merely out of curiosity. Wikipedia states "it is an adaptation of the Fell ...
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Projections in group $C^*$-algebras

Let $G$ be an amenable, discrete and infinite group. Cosinder its group C*-algebra $C^*(G)$ canonically represented on $B(\ell_2(G))$ by the left-regular representation $x\mapsto \delta_x$. Take the ...
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C* algebra of bounded Borel functions

Let $T\in B(H)$ is normal, and $B(\sigma(T))$ denote the $C^*$ algebra of all bounded Borel functions on $\sigma(T)$. Then is it true that $B(\sigma(T))$ is a closed $C^*$ algebra under the sup. norm ...
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Injectivity of index map for $K_1(S^1)$

This example/problem is from Valette's notes on the Baum-Connes conjecture (p. 45). The exercise is to prove that the (trivially equivariant) $K$-homology group $K_1(S^1)$ is $\mathbb{Z}$. For this, ...
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An strongly open set which is not measurable in the weak operator topology.

Let $H$ be a non-separable Hilbert space and $\{e_i\}_{i\in I}$ be an orthonormal basis for $H$. Let $J$ be a proper subset in $I$. Let us put $$E=\{x\in B(H): \lVert xe_j\rVert <1: j\in J\}$$ ...
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Relation of fundamental group $\pi_1(X,x_0)$ and properties of C(X)

As stated by commutative Gelfand Naimark theorem, every unital C* algebra is of the form C(X) for some compact Hausdorff space X. Moreover two such algebras are isomorphic iff the corresponding ...
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What is this operator topology?

Let $X$ be a separable Banach space with (norm $1$) Schauder-Basis $\{e_n\}_{n\in\mathbb N}$. Denote for $x\in X$ with $|\cdot|_x$ the seminorm on $\mathcal L(X)$ given by $|A|_x = \|A x\|$. Consider: ...
Let $H$ be a Hilbert space. We denote $K(H)$ by the space of compact operators on $H$ which is a two sided ideal in $B(H)$. Let $E$ be a norm closed convex subset of positive operators in $K(H)$ ...