# Questions tagged [operator-algebras]

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 of the tags (banach-algebras), (c-star-algebras), (von-neumann-algebras), and (operator-theory).

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### Any symmetric normed ideal $\mathfrak{a}$ on $\mathcal{H}$ is linearly generated by its positive elements

I have some questions about the proof of this statement in the book "Elements of Noncommutative Geometry" by Garcia-Bondía. A ideal $\mathfrak{a}\subset K(\mathcal{H})$ is called ...
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### Two projections are unitary equivalent and commute will give us self adjoint unitary equivalent?

I am reading Jones famous paper "index for subfactors" recently. And I met some questions in the reading. Here is the link for the paper:https://link.springer.com/article/10.1007/bf01389127. ...
1 vote
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### $C^*$-Algebras and Gelfand Duality Reference

I'm interested in learning more about the duality between (locally compact) Hausdorff spaces and commutative $C^*$-algebras. Does anyone have an introductory textbook they recommend? My background in ...
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### Why is the projection to the image of an algebra in the commutator

The proof in Murphy's book (chapter on Von Neumann algebras) of the fact that a *-subalgebra $A$ of $B(\mathcal{H})$ is strongly dense in $A''$. The proof procedes as follows: Take a $x\in \mathcal{H}$...
1 vote
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### If $\varphi : A\to B$ is surjective, then is $GL_{0}\left(B\right)$ contained in $\varphi\left(GL\left(A\right)\right)$?

Let $A$ and $B$ be two $C^*$-algebras. Suppose $\varphi : A\to B$ is a surjective $*$-homomorphism. Denote the set of invertible elements of $A$ by $GL\left(A\right)$ and denote the set of invertible ...
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### Von Neumann algebra decomposition as integral of factors and mixed state decomposition as sum of irreducible states

My question is the following. It is known that any Von Neumann algebra can be uniquely decomposed as integral over algebra factors. It is also know that any mixed state can be uniquely expressed as ...
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### Spectrum of a hermitian element of a C* algebra is connected?

In the proof of theorem 2.1.11 from C*-Algebras and Operator Theory, by Gerald Murphy the author is trying to prove $\sigma_B(u) = \sigma_A(u)$, where $B$ is a C*-subalgebra of $A$ and $u \in B$ is a ...
In "Disjointness preserving operators on $C^*$ algebras" by Manfred Wolff, Arch. Math 62, 248-253, 1994 it is presented the concept of zero divisor preserving map as follows: Let $A$...