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

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About GCR C*algebra

I cannot understand that a simple GCR C*algebra is *-isomorphic to the set of all compact operators on some Hilbert space. Please tell me how to show this.
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The left kernel of a positive linear functional and its $w^*$-extention

Let $A$ be a C*-algebra and $\phi$ be a positive linear functional on $A$. We let $\tilde{\phi}$ be its unique $w^*$-continuous extension on $A^{**}$. It is supposed to focus on the left kernel of ...
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A remark on projections in von Neumann algebras

Let $M$ be a von Neumann algebra and $e,f$ be projections in $M$. For a given central projection $z\in M$, is the following true? $$z(e\vee f)=ze\vee zf$$
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Restriction of a *-homomorphism

Let $A$ be C*-algebra then we know that $M_n(A)$ is also a C*-algebra. Let $\rho:M_n(A)\rightarrow B(K)$ be a *-representation of $M_n(A)$ on some Hilbert space $K$. Then there exists a ...
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Positivity in matrix algebra

Let $A$ be a unital C*-algebra. Then we know that $M_n(A)$ is also a C*-algebra. Let $x=[x_{ij}]\in M_n(A)$. I want to prove that if for every state $\phi$ on $A$ and for every ...
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Multiplicity free representation contain irreducible representation (for type I representation)?

While looking at Arveson's "An invitation to C* algebras", at the moment of defining type I representations (p. 47), he says that a (non degenerate) representation is type I if every central ...
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Which open projections are support of closed left ideals in a C*-algebra

Let $A$ be a C*-algebra. Let $\phi$ be a positive linear functional on $A$. The support of $\phi$ is defined by the smallest projection $q\in A^{**}$ with $\phi(q)=||\phi||$ and in the literature ...
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Idempotents which are not Mouray von neumann equivalent to its adjoint

What is an example of a $C^{*}$ algebra with an idempotent $e$ such that $e$ is not Mourray Von Neumann equivalent to $e^{*}$?
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States in a $C^*$-algebra bounded?

A functional $\phi$ on a $C^*$-algebra $A$ with unit element, i.e. $\phi: A \rightarrow \mathbb{C}$ is called a state if $\phi(T^*T) \ge 0$ for all $T \in A$ and $\phi( \operatorname{id}) = 1.$ Now, I ...
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the ideal structure of group $C^*$-algebras

What is the ideal structure of group $C^*$-algebras? Do there exist any books or articles in the field ? If G to be the group of integers $Z$ , then $C^*$($Z$)=C($T$). so because ideal structure of ...
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Support of a faithful representation

Let $A$ be a C*-algebra. Any representation $\pi:A\to B(H)$ is uniquely extended to a $w^*$-continuous representation $\tilde{\pi}:A^{**}\to B(H)$. Q: I am looking for an example of a faithful ...
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Is the bilinear map $M\times M^*\to M^*$ jointly continuous?

Let $M$ be a W*-algebra and consider the following map: $$\gamma: M\times M^*\to M^*: (a,f)\to af$$ where $af(b)=f(ba)$. Let us consider $M$ under the weak topology $\sigma(M,M^*)$ and $M^*$ under ...
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Independence of choice of faithful representation in reduced $C^*$ crossed product

In the definition of the reduced $C^*$ crossed product associated with an action of a discrete group $G$ on a $C^*$-algebra $A$, one can begin with any faithful representation of $A$ on a Hilbert ...
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Existence of uniform multiplicity projection in abelian Von Neumann algebras.

I am stuck in a proof in Davidson's "$C^*$ algebras by examples" book. In section II.3, he proves that any abelian Von Neumann algebra $N$ on a separable Hilbert $H$ has a non-zero projection with ...
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1answer
25 views

A normal positive linear functional whose support is 1

Let $M$ be a W*-algebra with unit $1$. Q1) Does there exist a normal positive linear functional $\phi$ on $M$ whose support is 1? Q2) We know that there is a unique central projection $z$ in ...
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the c*-algebra generated by the Volterra operator

Let V be the Volterra operator on $\mathscr{L^2(0,1)}$.$V(f)(x)=\int_{0}^{x}{f(y)dy}$. Show that $C^*(V)$, the smallest C* algebra generated with V, is $\mathbb{C}+\mathscr{B_0(L^2(0,1))}$ where ...
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A condition on quotient norms on quotient Banach algebras

Let $A$ be a non-unital Banach algebra, and let $A^+$ be the unitization of $A$ consisting of elements of the form $(a,z)$ where $a\in A$ and $z\in\mathbb{C}$ with multiplication ...
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Form of the Polar decomposition for $M_{\varphi}$

Polar Decomposition:Let ‎$‎‎v$ ‎be a‎ ‎continuous ‎linear ‎operator ‎on a‎ ‎Hilbert ‎space ‎‎$‎‎H$.then ‎there ‎is a‎ ‎uniqe ‎partial ‎isometry ‎‎$‎‎u\in B(H)$ ‎such ‎‎$‎‎v=u‎‎\mid ‎v‎\mid‎‎‎$ ‎and ...
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‎‎‎$‎‎C^*$-algebra generated by ‎$‎‎a$‎

Let ‎$‎‎A$ ‎be a unital ‎‎‎$‎‎C^*$-algebra. ‎‎ Assume that ‎$‎‎a\in A$ ‎is a ‎‎normal ‎and ‎invertible element ‎i.e ‎‎$‎‎aa^*=a^*a$ ‎and ‎‎$‎‎aa^{-1}=a^{-1}a=1$‎.‎ ‎let $‎‎C^*({a}) $ be the ...
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Why the set of pure state ‎is ‎weak* ‎compact?

Let ‎$‎‎A$ ‎be a‎ ‎C*-algebra‎. ‎ ‎$‎S(A)‎$ ‎is ‎the ‎set ‎of ‎state ‎on ‎‎$‎‎A$ and $‎‎PS(A)$ ‎is ‎the ‎set ‎of ‎pure ‎state ‎on ‎‎$‎‎A$. ‎ ‎ I ‎know ‎that ‎if ‎‎$‎‎A$ ‎is ‎unital ‎then ‎‎$‎‎S(A)$ ...
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‎Jointly ‎continuous of product in $B(H)$

‎Let ‎$‎B(H)‎$ ‎be the set of‎ ‎bounded ‎operators ‎on a Hilbert space ‎‎$‎‎H$.‎ ‎ I ‎know that ‎$u_{‎\alpha‎}‎\longrightarrow u‎$‎‎‎ ‎in ‎S.O.T ‎if ‎and ‎only ‎if‎ ‎$u_{‎\alpha‎}(x)‎\longrightarrow ...
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1answer
31 views

Is the square root function norm continuous?

Let $\{a_n\}$ be a sequence of positive operators in $B(H)$. What about the following implication, True or false? $$||a_n-a||\to 0\Longrightarrow ||a_n^{\frac{1}{2}}-a^{\frac{1}{2}}||\to0$$
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A remark on the polar decomposition

Let us focus on $L^1(H)$, the space of trace class operators on a Hilbert space $H$. Assume $\{x_n\}$ converges to $x$ (in the trace class norm). Let $x_n=u_n|x_n|$ and $x=u|x|$ be the polar ...
2
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1answer
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Essential ideals

I am trying to get my head around essential ideals. In literature I found 2 definitions: An ideal $I$ in a C*-algebra $A$ is essential in $A$ (i) if $aI = 0$ implies $a=0$, $a\in A$; or (ii) if ...
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postliminal $C^*$-algebra

A ‎$‎‎C^*$-algebra ‎‎$‎‎A$ ‎is ‎said ‎to ‎be ‎postliminal ‎if ‎for ‎every ‎non-zero ‎irreducible ‎representation ‎‎$‎(H,‎\varphi‎)‎$ ‎we ‎have ‎‎$‎‎K(H)‎\subseteq‎ ‎‎\varphi‎(A)‎$‎ ‎ In ‎Murrphy's ...
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Non spatial isomorphisms

Let $H$ be a Hilbert space. Any unitary operator $u\in B(H)$ induces an spatial isomorphism $\phi_u(x)=uxu^*$ on $B(H)$. Question: Let $\phi:B(H)\to B(H)$ be a surjective *-ismorphism. Is $\phi$ ...
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1answer
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Equivalent non-degenerate representations of C*-algebras

For two non-degenerate representations $\pi_j:A\to B(H_{\pi_j})$ ($j=1,2$), we write $\pi_1\sim\pi_2$ if there exists a $w^*$-continuous isometrically isomorphism from $\pi_1(A)''$ onto ...
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Normal positive functional on Von Neumann algebras

Let $A$ be Von Neumann algebra. A positive linear functional ‎$‎‎‎\varphi‎$ on $A$ ‎is ‎said ‎to ‎be ‎normal ‎if ‎for ‎any ‎self‎adjoint and increasing nets such that ‎$‎‎u_{\alpha‎}\longrightarrow ...
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C*-algebras: Proofs on $C_0(X)$

I'm looking to prove the following but am stuck, please can you help me? $C_0(X)$ is isomorphic as a C*-algebra to $C_0(Y)$ if and only if X is homeomorphic to Y, where X and Y are locally compact ...
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liminal and postliminal $C^*$-algebras

A $‎‎C^*$‎‎‎-algebra ‎$‎‎A$ ‎is said to be ‎postliminal (liminal) ‎‎ if for every non-zero irreducible representation‎$‎‎(H,‎\varphi)$ of ‎$‎‎A$ ‎we have‎‎‎ ‎‎$‎‎K(H)‎\subseteq ‎\varphi‎(A)$ ...
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Non-degenerate representations and central projections

Let $A$ be a C*-algebra and $\pi:A\to B(H)$ a non-degenerate *-representation. We also denote $M_{\pi}$ by the von Neumann algebra generated by $\pi(A)$ in $B(H)$. The predual of the von Neumann ...
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Show that the “folium” is norm closed

A C*-algebra $\mathfrak{A}$ is a Banach algebra with an involution operation $* : \left\lbrace\begin{aligned} \mathfrak{A} &\longrightarrow \mathfrak{A} \\ a &\longmapsto a^* \end{aligned} ...
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1answer
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closed convex hull of projection

$1$:I know that if ‎$‎‎F$ is a ‎locally convex ‎compact ‎space ‎then ‎‎$‎‎‎\overline{co}(‎Ext (F))=F$‎ ($Ext$: means extreme point) $2$:I ‎know ‎that ‎if ‎‎$‎‎M$ ‎is a ‎Von ‎Neumann ‎algebra ‎then ...
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1answer
24 views

projecton and positive element in $C^*$algebras

Let ‎$‎‎A$ ‎be ‎a‎ ‎$‎‎C^*$-algebra.‎$‎‎p\in A$ ‎is a ‎‎projections. ‎‎‎ Assume ‎that ‎‎$‎‎a$ ‎is a element in‎$‎‎ Ball(A_+)$ ‎such ‎that ‎‎$‎‎a‎\leq p‎$‎ Q:May I‎ ‎say ‎‎$‎‎ap=pa$?why?‎ ‎
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weakly convergent sequence of operator on $B(H)$

Let $H$ be a Hilbert space. Assume that $\{u_n\} \subseteq B(H)$ is W.O.T convergent.(‎ ‎$‎u_n‎\rightarrow u‎$‎‎ ‎in ‎W.O.T ‎topology ‎iff ‎‎$‎‎‎‎<‎u_n(x),y>\rightarrow‎ ‎‎‎<‎u_n(x),y> ‎ ...
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1answer
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*-isomorphism and spectrum

‎‎‎$A$ is a ‎‎‎‎$‎‎C^∗$-algebra and $P(A)$ is a set of projection of it. Assume that $A$ ‎admits a‎ ‎strictly ‎positive ‎element ‎‎‎‎‎$a$ ‎such ‎that ‎‎‎‎‎$‎‎‎‎σ(a)‎-\{‎0\}$ ‎is ‎discrete‎. I want to ...
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25 views

strictly positive elments $a$ when $‎‎‎\sigma(a) ‎\backslash‎ {0}‎$ ‎is ‎discrete

If ‎$‎‎A$ is a ‎‎$‎‎C^*$-algebra ‎and it ‎admits a‎ ‎strictly ‎positive ‎element ‎‎$‎‎a$ ‎such ‎that ‎‎$‎‎‎\sigma(a) ‎\backslash‎ {0}‎$ ‎is ‎discrete‎ then‎ Q1:‎$‎‎A$ admits ‎an ‎approximate ‎unit ...
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‎strictly ‎positive elements

Let ‎$‎‎A$ ‎be a ‎‎‎‎$‎‎C^*$-algebra‎. ‎$‎‎a\in A^+$ ‎is ‎strictly ‎positive in ‎$‎‎A$‎ ‎if ‎‎$‎‎‎\overline{aAa}=A‎$‎‎ *I know that if $A$ is unital, $a\in A^+$ is strictly positive iff $a\in ...
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2answers
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Example of an operator that is not subnormal

In some recent questions the term subnormal operator has appeared. A bounded operator $A$ acting on a Hilbert space $H$ is called subnormal if there exists a Hilbert space $K$ containing $H$ as a ...
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minimal subnormal extensions of $u$

Q1:Does every $u\in B(H)$ admit a minimal subnormal operator?why? Q2:Two minimal subnormal extensions of $u$ are equivalent.
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subnormal operator

I know that ‎$‎‎u\in B(H)$ ‎is a‎ ‎normal ‎operator if ‎‎$‎‎uu^*=u^*u$‎. I ‎know ‎that ‎if ‎‎$‎u‎$‎‎ ‎is ‎subnormal ‎‎‎‎then ‎‎‎ ‎‎$‎‎uu^*‎\neq ‎u^*u$ ‎(like unilateral shift operator). ‎‎ My ...
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1answer
79 views

Classification of representations of compact $C^*$ algebras for single operators.

I am looking at Arveson's book, an invitation to $C^*$ algebras. There, it is explained p. 21 ($C^*$ algebras of compact operators) that any representation of a compact $C^*$ algebra can be decomposed ...
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1answer
35 views

Maximal ideals of unital commutative $C^*$-algebra?

Define an ideal of a unital commutative $C^*$-algebra $A$ to be a proper subspace $I$ of $A$ such that $xy,yx\in I$ for all $x\in I$ and $y\in A$. Show that if $\lambda\in \widehat{A}$ (the ...
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an isomorphism from $L^\infty(\mathbb{T})$ to $L^\infty([-1,1],\displaystyle\frac{2}{\pi } \sqrt{1-t^2}\mathrm{d}t)$

$\mathbb{T}$ is the boundary of unit ball.Consier $\phi:[-1,1]\rightarrow\mathbb{T},\phi(t)=exp^{2i(arcsint+t\sqrt{1-t^2})},t\in[-1,1]$. It is easy to check that $L^2(\mathbb{T})\ni f\mapsto f\phi\in ...
2
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1answer
33 views

positive elements and norm

If $A$ is a abelian $C^∗$-algebra and $a,b$ are elements in $A$ such that $0‎≤‎a‎≤‎1,0‎≤‎b‎≤‎1‎‎$ ‎‎ then $0‎≤‎\|a-b \|≤‎1‎$. My problem is:"Does the same hold if $A$ is not abelian?"
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22 views

The absolute value of a bounded linear functional on a C*-algebra

Let us consider the commutative C*-algebra $C_0(\Omega)$. Let $\mu$ be a complex Radon measure on $\Omega$. By the Riesz representation theorem, $\mu$ may be considered as a bounded linear functional ...
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1answer
32 views

positive elements of $C^*$-algebra

If $A$ is a abelian $C^*$-algebra and $a,b$ are elements in $A$ such that $0‎\leq ‎a‎\leq ‎1,0‎\leq ‎b‎\leq ‎1‎‎$ then $0‎\leq ‎a‎b\leq ‎1‎$. My problem is:" Is it true if $A$ is not abelian?"
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1answer
52 views

States on a $C^*$-algebra

I know that if $A$ is a non-zero and unital $C^*$-algebra then $S(A)$ (the set of states on it) is weak${}^*$ compact. My problem is: Does the same hold if $A$ is not unital?
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36 views

Does this show boundedness of an operator?

In the second book on Operator Algebras and Quantum Statistical Mechanics by Bratteli and Robinson (on page 11) they are looking at the algebra of fermionic creation and annihilation operators ...
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1answer
38 views

Stinespring dialation

Let $A\in End(\mathbb{C}^n)$ be Hermitian and with positive eigenvalues. Define the linear map $T:End(\mathbb{C}^n)\longrightarrow End(\mathbb{C}^n)$ with $T(M)=(tr(M))A$. How can I make an isometry ...