# 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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### Equivalent definition of decomposable map

Let $A\subset B(H)$ be an operator system and $B$ be a $C^*$-algebra. (1)$u:A\rightarrow B$ is called a decomposable map if $u$ is in the linear span of $CP(A, B)$, where $CP(A,B)$ is the set of all ...
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### Construction of the free product of $C^*$-algebras

The content of the screenshot is from Pisier's book. I have two questions ： $\mathcal{F}$ is an algebra, how to conclude that $\pi: \mathcal{F} \to B(H_{\pi})$ is a $*$-homomorphism? 2.Why is $j$ ...
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### Is the center $Z$ of a C*-algebra $A$ unital if and only if $A$ is unital?

Let $A$ be a C*-algebra. Its center is defined by $$Z:=\{z\in A:az=za \ \text{for all} \ a\in A\}.$$ It is easy to verify that $Z$ is a C*-subalgebra of $A$. Also, if $A$ is unital, then it is clear ...
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### Is a representation $(H,\phi)$ of a simple C*-algebra $A$ always faithful?

Suppose that $A$ is a simple C*-algebra (i.e. there is no closed ideal $I\subset A$ such that $0\neq I\neq A$) and let $(H,\phi)$ be a representation. Can we conclude that $(H,\phi)$ is faithful, i.e. ...
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### Prove the following are equivalent with irreducibility of representation on $C^*$-algebras

Let $\pi: A \to B(\mathcal{H})$ be a representation where $A$ is a $C^*$-algebra and $\mathcal{H}$ a Hilbert space. I'm trying to show the following four statements are equivalent: (1) $\pi$ is ...
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### Set of representations of a $C^*$-algebra

I saw the following statement from a reference book. Let $S$ be the set of representations of a $C^*$-algebra $A$. Does it mean that $S$ is the set of unitary equivalence classes of the ...
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### If $(H,\pi)$ is a finite dimensional irreducible representation of a C*-algebra $A$, then $\pi(A)=B(H)$

Let $A$ be a C*-algebra and $(H,\pi)$ an irreducible representation such that $\dim(H)<\infty$. Then how do I show that $\pi(A)=B(H)$? Here is what I tried: Since $(H,\pi)$ is irreducible, we ...
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### $*$-isomorphisms of cross product $C^*$-algebras

Let $G$ be a discrete countable group, $B$ is the CAR algebra $\otimes_{\Bbb N}M_2$. I feel confused about the statement marked green in the screenshot. How to construct the explicit bijections?
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### Redundant assumption in proving the quotient of a Banach algebra over a modular ideal is a field

When reading Murphy's book $C^*$ algebras and operator theory, I found the Lemma 1.3.2 a bit strange. The original lemmas is as the following: If $I$ is a modular maximal ideal of a unital abelian ...
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### Infinite tensor product

Suppose that $X$ is an infinite set and $A$ is a unital $C^*$-algebra. The tensor product $\bigotimes_X A$ is defined to be the closed linear span of $\bigotimes_{x\in X }a_x$, where $a_x\in A$ for ...
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### Involution is not strongly continuous

Let $H$ be a Hilbert space and consider on $B(H)$ the strong topology, i.e. the topology induced by the seminorms $$x \mapsto \Vert x \xi \Vert, \quad \xi \in H$$ I.e. a net $(x_\alpha)_\alpha$ ...
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### Univesal Coefficient Theorem for $C^*$-algebras

The UCT theorem is shown in the sreenshot. I have a question : What is the definition of $Ext_{\Bbb Z}^1(K_{*}(A),K_{*}(B))$? Does it have a relationship with Tor functor?
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### $C^*$ algebras and differential geometry

If $(A, G, \alpha)$ is a $C^*$-dynamical system where $G$ is a Lie group and $\alpha: G \rightarrow \operatorname{Aut}(A)$ is a continuous homomorphism (where $\operatorname{Aut}(A)$ is equipped with ...
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### Are bounded operators on a Banach space a $C^*$-algebra?

It is well-know that if $H$ is a complex Hilbert space then the set of all bounded operators $\mathcal{B}(H)$ is a $C^*$-algebra. If we change $H$ with a complex Banach space $E$, do we have the same ...
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### Gauss divergence theorem applied to operator valued functions

I'm studing quantum field theory. Especifically the procedure called second quantization for the complex scalar field. I noticed that I can derive the Klein Gordon equation from the Heisemberg ...
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### Bott-Periodicity for unitary matrices with entries in general $C^{*}$-algebra

When working through the proof of Bott-Periodicity (the original proof by Bott), I noticed that compactness of $U(n)$ is important as it gives us that every geodesic looks like an exponential map (...
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### Every *-isometric isomorphism of $B(\mathcal{H})$ keep compact operators?

Let $\mathcal{H}$ be a Hilbert space, $B(\mathcal{H})$ denotes the $\mathcal{C}^*$-algebra consisting of bounded linear transformation on $\mathcal{H}$ ($*$ is the adjoint). Now consider a isometric *-...
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### Simple $C^*$-algebras are not commutative

Is it true that for simple $C^*$-algebras, meaning that they don't have non-trivial two-sided ideals, it holds that they are necessarily non-commutative? And why?
### $C^*$- systems with positive maps
Which is the main reason of defining a $C^*$- system as a pair of a $C^*$- algebra and a (unital) positive map $\Phi : \mathfrak{A} \rightarrow \mathfrak{A}$ ? What is the main reason behind choosing ...
### Algebra product on the commutative $C^*$-algebra $\mathbb{C}^N$
If we think of $\mathbb{C}^N$ as a commutative $C^*$-algebra, then it will be the algebra of continuous functions on $X=\{1,2, \dots, N\}$. So the only functions that are continuous are constants, (do ...