# Tagged Questions

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### Completely positive maps-equivalent definition

The most usual definition of the completely positive map (c.p.) between two C*-algebras (say, unital) is the following: $\sigma: A \to B$ should satisfy $\sigma(1)=1$ and for each $n \in \mathbb{N}$ ...
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### Invertible operator with countable spectrum

Let $H$ be a separable Hilbert space and $A$ is an invertible bounded operator on $H$. Can we approximate $A$ with an invertible operator $B$ such that $sp(B)$ is a countable set? Motivation: ...
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### ${\rm II}_1$-factors with finite commutant and trivial intersection generate $B(H)$?

Let $H$ be an $\infty$-dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Let $\mathcal{A}$, $\mathcal{B} \subset B(H)$ be ${\rm II}_1$-factors such that $\mathcal{A}'$, ...
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### How much does the absolute value of an operator behave like an absolute value?

Recall that the absolute value of a bounded operator $T$ on a Hilbert space $H$ is the unique positive operator $|T|$ such that $$\||T|x\|=\|Tx\|$$ for all $x\in H$. It can be defined using the ...
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### “Minimal” group C*-algebra?

Let $\Gamma$ be a discrete group (though this could be asked for general locally compact groups) and consider the Banach $*$-algebra $\ell^1(\Gamma)$. We have two natural $C^*$-algebra completions: ...
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### Characterization of ideals in the bounded operators

Let $\mathcal{B}(H)$ denote the C*-algebra of all bounded operators on a separable infinite dimensional complex Hilbert space $H$. It is a well-known fact that $\mathcal{B}_0(H)$, the ideal of compact ...
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### Why do the projections in the Calkin algebra not form a lattice?

Let $H$ be an infinite dimensional separable complex Hilbert space. Denote by $\mathcal{B}(H)$ the C*-algebra of bounded operators on $H$, $\mathcal{K}(H)$ the ideal of compact operators on $H$, and ...
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### Continuity of the product map

Let $A$ be a $C^*$-algebra. Is it possible to characterize $A$ for which the product map defined by $$\sum\limits_{i=1}^n a_i\otimes b_i \mapsto \sum\limits_{i=1}^n a_i b_i$$ is continuous with ...
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### Continuous linear functionals in strong operator and $\sigma$-strong topologies

It was mentioned in the comments to http://math.stackexchange.com/questions/517369/comparison-of-strong-operator-and-weak-topologies-on-bh that continuous linear functionals on ...
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### Differentiable Path of Operators and their Inverses

Let $\mathcal{H}$ be a separable Hilbert space. Consider a differentiable map $\mathbb{R} \rightarrow \mathcal{B}(\mathcal{H}), t \mapsto A(t)$, where $\mathcal{B}(\mathcal{H})$ is the space of ...
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### How many ideals are there in $B(H)^{**}$?

It is well-known (and easy to prove) that the only closed ideals of $B(\ell_2)$ are $\{0\}$, $B(\ell_2)$ and $K(\ell_2)$, the ideal of compact operators on $\ell_2$. I am curious whether we know what ...
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### Which Sigma-Ideals in a Sigma-Algebra are Ideals of Null Sets?

My question is motivated, to be somewhat vague, by an attempt to see how much a measure space is defined by the set of null sets. In other words, assume we are not given a concrete measure on a space ...
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### K-Theory of Algebra of Zeroth Order Pseudo differential operators

Any one knows a reference for computing K_0 of Algebra of zeroth order Pseudo's on a closed manifold in terms of explicit generators? Thanx!
Let $A$ be a Banach algebra with the following property: For every two nets $x_{\alpha}$ and $y_{\alpha}$ in $A$, $x_{\alpha}y_{\alpha}$ converges if and only if $y_{\alpha}x_{\alpha}$ converges. ...