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).

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19 views

Compute the norm of the sum of two bounded linear functionals

Let $\omega_1$ and $\omega_2$ be two states on a $C^*$-algebra $A$. Set $\tau=\omega_1+i\omega_2$. Can we get the following conclusion: $\|\tau\|=\sqrt{2}$ if and only if $\omega_1=\omega_2$?
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34 views

Representations of a semigroup over a Hilbert space

A representations of a discrete semigroup $S$ with an involution $\star$ over a Hilbert space $H$ is a semigroup homomorphism $\varphi : S \to B(H)$ that preserve the involution. That is, for any $a, ...
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24 views

GNS construction for a normal state on a von Neuamann algebra

Let $am$ be a von Neumann algebra. For any faithful normal state $\omega$ on $M$, according to the GNS construction, we have $\omega(x)=\langle x \Omega,\Omega\rangle$, where $\Omega$ is a cyclic ...
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GNS constructions of two faithful positive linear functional

If $\rho_1$ and $\rho_2$ are two faithful positive linear functionals on a unital $C^*$-algebra $A$. Then $\rho_i(x)=\langle x\omega_i,\omega_i \rangle$, where $\omega_i=1+N_{\rho_i}$, $N_{\rho_i}=\{x\...
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36 views

Unitization of a $C^{*}$-algebra - completeness of the constructed norm

I have a question about the unitization of $C^{*}$-algebras. More precisely, a question about the proof of the following statement: If $A$ is a (possibly unital) $C^{*}$-algebra, then there is a ...
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27 views

Show that ($C(S)$ weakly complete) $\implies$ $S$ finite

$\hspace{0.44cm}$As the title says, I am trying to show that given a compact Hausdorff topological space $(S, \tau)$, if $C(S)$ is weakly complete then $S$ is a finite set (i.e. $C(S)$ is finite-...
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16 views

Constructing a proper dense subalgebra from a subalgebra inside an ideal

Given a C* algebra $A$ and *-sub-algebra $B$, where $B$ is contained in a two-sided, closed ideal $I$, can we construct a proper, dense sub-algebra that contain $B$? I am considering the following ...
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23 views

Proving that $A\oplus\mathbb{C}\ni(a,\lambda)\mapsto L_{a}+\lambda I\in B(A)$ is injective

Let $A$ be a non-unital $C^{\star}$-algebra and let $B(A)$ be the Banach algebra of bounded linear operators from $A$ to itself. For $a\in A$ we consider the left-multiplier $L_{a}\colon A\to A$ ...
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37 views

Dense subalgebra intersection with essential ideal is dense?

Consider $A$ is an unital C* algebra, $D$ is a dense *-subalgebra, $B$ is a essential,maximal ideal of $A$. Is $D\cap B$ dense in $B$? I have seen this question, and this question, but my assumption ...
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29 views

why $\tilde{A}/A$ is isomorphic to $\mathbb{C}$?

When I am reading the answer of this question, I found the statement $\tilde{A}/A\simeq \mathbb{C}$ suspicious to me. Here $A$ is a non-unital C* algebra and $\tilde{A}$ is it unitization. I don't ...
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44 views

sub-algebra of inductive limit(II)

This question is similar to my question here, but not the same question. Let $(A_{i},\alpha_{i})$ be directed system of C* algebras and *-homomorphisms. Let the $\beta_{i}:A_{i}\rightarrow A$ are the ...
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33 views

What is the smallest dimension a non-commutative C*-algebra can have?

What is the smallest dimension a non-commutative C-star-algebra can have? Let $d$ denote this dimension. Clearly, $d\leq 4$ as $M_{2}(\mathbb{C})$ is a $4$-dimensional non-commutative C-star-algebra. ...
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Characterization of semi-finiteness in terms of the modular automorphism group

The Theorem in the screenshit is from Takesaki's book (Vol 2, Chapter VIII, section 3) When reading the proof of (iii)→(i), I met with troubles. How to prove $n_{\varphi}$ is a left ideal according ...
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36 views

Trying to understand definition of Lie ideal for C*-algebras

Let $A$ be a $C^*$-algebra. A sub space $I$ of $A$ is called Lie ideal of A if $[I,A]= IA-AI \subset I$ Since I contains $0$, isn’t it this definition equivalent to definition of two sided ideal of $...
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1answer
28 views

Dense subalgebras of the unitization and their intersection with the original $C^*$-algebra

As I was trying to figure out this question, I wondered if the following is true: If $A$ is a non-unital $C^*$-algebra and $\tilde{A}$ denotes its unitization, then is it true that any $*$-subalgebra $...
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59 views

multiplier algebra and intersection

Let $M(A)$ be the multiplier algebra of non-unital C* algebra $A$. So $A$ is essential ideal of $M(A)$. Does there exists a dense $\ast$-subalgebra $B$ of $M(A)$, such that $B\not\subset A$, and $B\...
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36 views

functional calculus on C*-algebra and majoration of $||f(A+B)-f(A)||$

In introductory books on C*-algebra, functionnal calculus is quicly presented as it is a powerful tool in the field. It is for example prooved that for an operator $a$ and a function $f$ continous on ...
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25 views

Check a set is a von Neumann subalgebra

Let $M$ be a von Neumann algebra and $\varphi$ is a fithful normal positive linear functional on $M$. Define $S:=\{x\in M: x\varphi=\varphi x\}$, where $x\varphi(y):=\varphi(yx),\varphi x(y):=\varphi(...
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34 views

Faithful semi-finite normal weights

The above screenshot is from Takesaki's book(Vol 2, Chapter VIII) I wonder how to prove the implication (ii)→(i)? By Theorem 2.11, we have $\sigma_t^{\psi}(x)=h^{-it}\sigma_t^{\varphi}(x)h^{it}$ for ...
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Functional calculus - Typo in Blackadar's Operator Algebras?

I was reviewing some C*-algebra theory in Bruce Blackadar's Operator Algebras - Theory of C*-Algebras and von Neumann Algebras, when I came upon what seems to be a typo. On page 61, the following ...
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55 views

State on *-algebra: $\phi(|f+g|^2)$ when $\phi(|f|^2),\phi(|g|^2)>0$ for monomials

Let $\mathcal{A}$ be the unital *-algebra generated by $N^2$ projections $u_{i,j}=u_{i,j}^2=u_{i,j}^*$, such that the rows/columns are partitions of unity $\sum_k u_{ik}=1_\mathcal{A}=\sum_k u_{kj}$, ...
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Equivalence of the two definitions of multiplier algebra.

$M(A)=\{(L,R)\in B(A)\times B(A):aL(b)=R(a)b\ \text{for all }a,b\in A \}$, where $B(A)$ are the bounded linear operators on $A$. $M(A)=\{x\in A'': xA,Ax\subseteq A\}$, where $A''$ is the enveloping ...
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51 views

Factor a rank one operator on a Hilbert C*-module.

Let $A \subset \mathcal{L}(\mathcal{H}_0)$ be a concrete $C^*$-algebra and $X$ be a right Hilbert A-module. For each $x,y \in X$ we have rank one operators \begin{align} \theta_{x,y}: X & \to X \\ ...
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53 views

Pure states on noncommutative $\mathrm{C}*$-algebra can “be disturbed”

In a finite dimensional commutative $\mathrm{C}^*$-algebra $A\subset B(\mathbb{C}^N)$, if a vector state $\varphi_x(f)=\langle x,f(x)\rangle$ given by $x\in P(\mathbb{C}^N)$ is pure, then there is no ...
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43 views

Isomorphism $(V \otimes W)^* \cong V^* \otimes W^*$`

Let $A$ be a reflexive $C^*$-algebra. We have a multiplication map $A \otimes A^* \rightarrow A^*$, i.e. $A^*$ is a $A$-module. We get a map $A^{**} \rightarrow (A \otimes A^*)^*$. Is it possible to ...
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45 views

Sum of two positive elements is positive in a $C^*$ algebra is positive

I think this is easy to see, using Gelfand Transform. Using the transform we can see $spec(a)=\{\phi(a)| \phi\in Spec(A)\}$ thus $spec(a+b)=\{\phi(a)+\phi(b)|\phi\in Spec(A)\}$ Now if both $a$ and $b$ ...
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47 views

Dixmier's explanation of why $L^1(G)$ is not a $C^*$-algebra

I've recently been learning some basic facts about $C^*$-algebras and their connections to representations of locally compact groups $G$, and I'm currently trying to understand the definition of the $...
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Support Projection for Characters in $\mathrm{C}^*$-Algebra with finitely many characters

Related to this question... For a normal state $\varphi$ on a von Neumann algebra $\mathcal{M}$ there is a projection $p_\varphi$ with some nice properties called its support. Some properties include ...
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33 views

In a $C^*$ algebra why $\|a\|\not =\rho(a)$ for any $a$?

In a $C^*$ algebra why $\|a\|\not =\rho(a)$ for any $a$? Where $\rho(a)$ is the spectral radius. It can be shown that the equality holds for self-adjoint elements. Then that can be used to show that ...
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31 views

Faithful state and associated measure.

Let $A= C^*(1,a)$ where $a \geq 0$, that is $A$ is a unital $C^*$-algebra generated by a positive element $a$ and the identity. Let $f: A \to \mathbb{C}$ be a state on $A$. We have a canonical ...
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51 views

Is an exact-by-nuclear extension of $C^*$-algebras again exact?

I know that in general exact-by-exact extensions of $C^*$-algebras need not be exact. Is it true that, if we have a short exact sequence of $C^*$-algebras $$0 \to I \to A \to B \to 0$$ such that $I$ ...
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34 views

Special case of Elliott's Theorem

Let $A$ and $B$ be unital $AF$-algebra. By Elliott's theorem we know that if there an order isomorphism $\psi: K_0(A) \rightarrow K_0(B)$ with $\psi([1_{A}]) = [1_{B}]$, then there exists an ...
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58 views

Tracial topological rank of hereditary

Let $A$ be a unital simple $C^*$-algebra with $TR(A) \leq k$ (tracial topological rank). I can prove that for any unital hereditary $C^*$-subalgebra $B$ of $A$, $TR(B) \leq k$. If $TR(A) \nleq k - 1$, ...
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50 views

Embedding of separable $C^*$-algebra

Let $A$ be a separable $C^*$-algebra. Can $A$ embedded into a separable, simple $C^*$-algebra?
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Quotients of finite C-star algebras: are they finite?

Edit: to whomever downvoted this post: please explain why. Seriously, I'd like to know why because I cannot think of any reason. This is a fine question, it got 3 upvotes, in my humble opinion it is ...
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1answer
69 views

C*-Algebras and Operator Theory - Gerald J. Murphy Theorem 4.1.2 [closed]

I'm stuck On the theoremTheorem 4.1.2, from C*-Algebras and Operator Theory by Gerald J. Murphy 4.1.2. Theorem. Suppose that $(p_\lambda)_{\lambda\in\Lambda}$ is a net of projections on a Hilbert ...
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Does a $\mathrm{C}^*$-algebra generated by projections contain support projections

I think we have for a (normal?) state $\varphi$ on a von Neumann algebra $\mathcal{M}$ a projection $p_\varphi$ with some nice properties called its support. It arises as follows: Define the following ...
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1answer
38 views

Operators embedded in second dual as continuous functions on state spaces

Let $A$ be a unital $\mathrm{C}^*$-algebra. There are some well-known facts about the state space $S(A)$ and the set of pure states $PS(A)$. We might remark that in general $PS(A)$ is not weak$*$-...
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39 views

Extending a pre-Hilbert module operator over a pre-$C^*$-algebra to a Hilbert module over the completion

Let $A_0$ be a pre-$C^*$-algebra, i.e. $A_0$ satisfies all the criteria of a $C^*$-algebra except that it need not be complete. Then we can view $A_0$ as an inner product module over itself, i.e. a $\...
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56 views

Invertible positive elements

In a $C^*$-algebra ${\cal A}$, I know that $a\in {\cal A}_+$ if and only if $a=x^*x$ for some $x\in {\cal A}$. Question: If we know that $a$ is also invertible, can we choose $x$ to be invertible? It ...
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37 views

$\mathcal A$ is a Banach algebra. Can there be two different $*$ operators which both make $\mathcal A$ a $C^*$-algebra?

I really have no idea where to start. The only thing I know is that there can not be different $C^*$-norms (whether complete or not) on a $C^*$-algebra, but I find that I barely know nothing about ...
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38 views

Intuition for universal pair of unitaries and group $C^*$-algebra

Let $F_2$ be the free group on two generators. If $\pi$ is the universal unitary representation of $F_2$ on a Hilbert space ${\cal H}$, then the group $C^*$-algebra $C^*(F_2)$ is the $C^*$-subalgebra ...
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1answer
39 views

Ultraweak convergence on matrix over a von Neumann algebra

Let $N$ be a von Neumann algebra and $n\geq1$be an integer. Let $x_\lambda=[x_{i, j}^\lambda]_{i,j} $ be a net in $M_n(N)$. Is it true that $(x_\lambda)_\lambda$ converges ultraweakly to zero in $M_n(...
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55 views

Pure State Definition Confusion

Let me start with some objects. Consider the $\mathrm{C}^*$-algebra $A$ defined by: $$A=M_1(\mathbb{C})\oplus M_2(\mathbb{C})\subset B(\mathbb{C}^3).$$ Let $x=\mathbb{C}^3$ be given by $(e_1+e_2)/\...
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1answer
39 views

Spectral Theory for “Unrepresented” C*-algebras

If you Google something like "Borel functional calculus" or "spectral decomposition" you get plenty of results for operators on Hilbert spaces. What about an (unrepresented) $\...
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54 views

$C^*$ algebra of LCA-group

I'm reading A. Deitmar and S. Echterhoff's book "Principles of Harmonic Analysis", and have a confusion of the proof in the snapshot: Why do we need to prove $\|f\|\equiv\|f\eta\|$ in order ...
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1answer
37 views

Multiplier algebra of a $C^*$-subalgebra of $\mathcal{B}(H)$ [closed]

Let $A$ be a $C^*$-subalgebra of $\mathcal{B}(H)$ for some Hilbert space $H$. Question: Is the multiplier algebra of $A$ $*$-isomorphic to some $C^*$-subalgebra of $\mathcal{B}(H)$?
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22 views

An exercise in Takesaki's book (Vol2, Chapter VIII)

I was confuses abot the third problem. 1.If $\varphi\in M_{*}^+$ and $\varphi$ is faithful, why could we have the modular automorphism group $\sigma_t^{\varphi}$. According to the definition of ...
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1answer
82 views

Technical lemma about state space of a $C^*$-algebra.

Consider the following proof from the book "$C^*$-algebras and finite-dimensional approximations": Why does this proof work in the non-unital case? (see the last line). Maybe we have $$\|a +...
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1answer
24 views

$C^*$-algebraic tensor product preserves strict inclusions?

Let $I$ be a proper closed ideal of a $C^*$-algebra $A$ and let $B$ be a unital (for simplicity) $C^*$-algebra. We have a natural inclusion $$I \otimes B \subseteq A \otimes B$$ Here, the tensor ...

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