# Questions tagged [c-star-algebras]

A C*-algebra is a complex Banach algebra together with an isometric antilinear involution satisfying $(ab)^*=b^*a^*$ and the C*-identity $\Vert a^*a\Vert=\Vert a\Vert^2$. Related tags: (banach-algebras), (von-neumann-algebras), (operator-algebras), (spectral-theory).

1,572 questions
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### A sequence of points in the spectrum of a subhomogeneous C$^{*}$-algebra can converge to at most finitely many points

Let $A$ be a subhomogeneous C$^{*}$-algebra (i.e., there is a finite upper bound on the size of the irreducible representations of $A$). Let $\hat{A}$ denote its spectrum. I heard of a result that ...
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### sum of ideals in $C^*$ algebra

Suppose $I_1$ is a maximal ideal in $C^*$ algebra $A$,$I_2$ is an ideal of $A$,then $I_1+I_2$ is an ideal of $A$,can we conclude that $I_2\subset I_1$? My thought:if there exists an element $x\in I_2$...
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### Universal $C^*$ algebra subject to $x^*x+y^*y=xx^*+yy^*$

What is the precise description for the universal unital $C^*$ algebra generated by two elements $x,y$ subject to relation $$x^*x+y^*y=xx^*+yy^*$$
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### positive functionals on the full group C*-algebra

It it true that every positive linear functional on the full group C*-algebra is Completely positive? I am reading Brown and Ozawa's book and they seem to use this at some point, yet I'm not sure how ...
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### Spectrum of the difference of almost commuting elements

Suppose $A$ is a C*-algebra. Suppose we have an approximate unit $(u_\lambda)$ then one knows that $|| u_\lambda a-au_\lambda||\rightarrow 0$ in particular $u_\lambda$ almost commutes with $a$ can ...
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### find a smallest nuclear $C^*$ algebra containing set S [closed]

Suppose $S$ is a set ,can we find a smallest nuclear $C^*$ algebra containing $S$
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### Bounded linear functional on a $C^*$ algebra

I want to prove the following statement: Let $\mathcal{A}$ be a unital $C^*$ algebra with unit $e_\mathcal{A}$ and let $\varphi \in \mathcal{A}^*$ be a bounded linear functional in $\mathcal{A}$ ...
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### stably finite $C^*$ algebras

I saw a conlusion:If $A$ is simple $C^*$ algebra,then $A\otimes \Bbb K$ contains no infinite projections? If $A$ is simple, $A\otimes \Bbb K$ contains no infinite projections,can we conclude that $A$...
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### construct a representation of a $C^*$ algebra

If $A$ has a tracial state,we can construct a representation $(\pi,H)$ by the GNS theorem. My question is: If $A$ has no tracial states,how can we construct a representation of $A$?Do there exist ...
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### Von Neumann algebra generated by a set

Suppose $A$ is a unital $C^*$ algebra,$\pi:A\to B(H)$ is a representation of $A$.Then the von Neumann algebra generated by $\pi(A)$ is equal to $\pi(A)^{"}$. Is the weak$*$ closure of $\pi(A)$ equal ...
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### invariant subspace of a representation

Suppose $\pi:A \to B(H)$ is a representation of $A$ such that $\pi(A)K_1\subset K_,\pi(A)K_2\subset K_2$,where $H=K_1\oplus K_2$,can we conclude that there exist a projection $p\in \pi(A)^{'}$ such ...
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### Decomposition of self adjoint elements by positive elements

Let $a \in A$ be a self adjoint element of a $C^*$ algebra. There exists positive elements $a_+, a_-$, such that $$a=a_+ - a_{-}$$ $$a_+a_-=a_-a_+=0$$ Is the statement true? This is ...
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### Showing isomorphism of two $C^*$ algebras

It seems that quite a standard trick of showing two $C^*$ algebras are as follows: Let $A$ be a $C^*$ algebra $B$ another $C^*$ algebra. $A' \subseteq A$ be a subalgebra that is closed under $*$. (...
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### Maximal abelian subalgebra in generated von Neumann algebra

Let $D \subseteq A$ be an abelian C*-subalgebra of the C*-algebra $A$ where $A \subseteq B(H)$ for some separable Hilbert space $H$. Assume that the von Neumann Algebra generated by $D$ is a maximal ...
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### Showing a C* algebra with certain properties has a minimal projection

I am trying to show the following which is stated in Exercise 10.11.10 of Blackadars book on K-theory for operator algebras. A unital, simple, nuclear, stably finite, infinite dimensional C*-algebra ...
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### Hilbert C*-Modules: Inner *-Isomorphisms

I have got a very basic question, but it would simplify some things, so I hope this resolves in either the affirmative or maybe someone can provide an I guess simple non-example. Given pairs of ...
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### Maximal tensor product of quotient C*-algebras

This question is from the book: C*-algebras and Finite-Dimensional Approximations by N.P.Brown and N. Ozawa Ex 13.3.5. Let C$_i$ (i=1,2) be C*-algebras with the LLP and J$_i$ be a closed two-...
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### $f+g$ $*$-homomorphsim if and only if $im \, f \cdot im\, g = 0$

Let $f,g:A \rightarrow B$ be $*$-homomorphisms of $C^*$ algebras. Then $f+g$ is a $*$-homomorphism if and only if $im \, f \cdot im \, g =0$. How does this hold? My thoughts: We know that $f+g$ is ...
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### multiplier algebra of a non-unital $C^*$ algebra

Given any infinite dimensional unital $C^*$ algebra $A$,does there must exist a non-unital $C^*$ algebra $B$ such that the multiplier algebra $M(B)$ of $B$ is $A$?
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### Is the set of normal, positive, faithful, linear functionals on a W*-algebra open?

Let $\mathcal{A}$ be an infinite-dimensional W*-algebra, that is, an infinite-dimensional $C^{*}$-algebra which is the Banach dual of a Banach space $\mathcal{A}_{*}$ (equivalently, $\mathcal{A}$ is ...
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