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 ...

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Double dual of the space $C[0,1]$

The second dual or double dual of the space of all continuous functions on $[0,1]$, $C[0,1]$ is von Neumann algebra. Can anyone help me identifying this space?
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1answer
19 views

Jones construction projections

Let M be a von Neumann algebra with faithful normal normalized trace tr. Let $\{ e_i | i=1,2,\dots \}$ be projections in M such that: $e_ie_{i \pm 1}e_i=\tau e_i $ for some $\tau \leq 1$ ...
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votes
1answer
44 views

support of an operator on a Hilbert space

Let $x\colon\mathcal{H}\to\mathcal{H}$ be a self-adjoint operator, the support $s(x)$ of $x$ is defined as the smallest projection $e\in B(\mathcal{H})$ such that $ex=xe=x$. Let $x=\int\lambda \, ...
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16 views

$p$-complete boundedness of homomorphisms between $L^p$ operator algebras

Let $A$ and $B$ be non-unital Banach subalgebras of $B(L^p(X,\mu))$ where $p\in[1,\infty)$. We unitize $A$ (and similarly for $B$) by considering $\tilde{A}=A+\mathbb{C}I\subset B(L^p(X,\mu))$, and we ...
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24 views
+50

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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9 views

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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25 views
+50

Finite dimensional Banach algebras whose $K_{0}$ group is a non trivial finite group

Motivated by this question we ask Is there a finite dimensional Banach algebra $A$ such that $K_{0}(A)$ is a nontrivial finite group? I understand from the above link and this post that any ...
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votes
1answer
39 views

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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14 views

Norm convergence of a net of operators

Let $T$ be a positive operator in B(H). For every $\epsilon >0$, define $T_{\epsilon}:=(T+\epsilon I)^{-\frac{1}{2}}$. This makes sense since the spectrum of $T$ lies in $[\epsilon,\infty)$. Let ...
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1answer
25 views

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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14 views

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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1answer
21 views

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 ...
3
votes
1answer
19 views

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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0answers
16 views

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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0answers
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Relationship between group $C^\ast$-algebras $C^\ast(G)$ and graph $C^\ast$-algebras $C^\ast(E)$

Let $E=E(G,S)$ be the graph defined by a group $G$ and a subset $S$ of $G$. What is relationship between group $C^\ast$-algebras $C^\ast(G)$ and graph $C^\ast$-algebras $C^\ast(E)$?
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1answer
20 views

Minimum of the Schatten 1-norm

Given two operators or non-zero matrices $A$ and $B$, where $A\neq B$, tr$(A)=1$ and tr$(B)=1$ and tr$(A-B)=0$, what is a lower bound of the Schatten p-norm ($p=1$) $\|A-B\|_1$? Any helpful ...
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22 views

Structure of crossed product $C(X)\rtimes_r G$ when $G$ is locally finite

Let $G$ be a discrete group acting on a compact Hausdorff space $X$. Then we may consider the reduced crossed product $C^*$-algebra $C(X)\rtimes_r G$. When $G$ is finite, I know that we essentially ...
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0answers
33 views

example of positive operators a,b, $a\le b$ but $b^2-a^2$ is not positive [closed]

Give an example of a C*-algebra $\mathscr{A}$ and positive elements a,b in $\mathscr{A}$ such that $a\le b$ but $b^2-a^2\notin \mathscr{A}_+$, i.e. $b^2-a^2$ is not positive element in $\mathscr{A} $
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6 views

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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0answers
23 views

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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0answers
15 views

the c*-algebra generated by a closed ideal and a c*-subalgebra

If $\mathscr{A}$ is an unital c*-algebra, $I$ is a closed ideal of $\mathscr{A}$ ,and $\mathscr{B}$ is a unital c*-subalgebra of $\mathscr{A}$ . Show that the c*-algebra generated by ...
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0answers
17 views

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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1answer
250 views

Comparison of Strong OPerator and Weak * Topologies on B(H)

It is known that in $\mathfrak{B}(\mathbb{H})$, the weak operator topology (WOT) is contained in both the strong operator topology (SOT) and $\sigma$-weak topology. In general the SOT and the ...
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votes
1answer
77 views

Commuting nets for commuting projections

Let $A$ be a $C$*-algebra and $p,q\in A^{**}$ be commuting projections. Then there exist self-adjoint nets $(x_i)_i$ and $(y_j)_j$ in $A$ with $x_i\to p$ and $y_j\to q$ in the weak *-topology. Can ...
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1answer
22 views

Closed ideals in $\mathbb B(H)$

Let $\mathbb{H}$ be a non-separable Hilbert space. If $\alpha$ is an countably many infinite cardinal number, let $I_{\alpha}=\{A\in \mathbb{B(H)}\:dim~ cl(ran A)\le \alpha\}.$ Show that ...
18
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1answer
719 views

Motivation for abstract harmonic analysis

I am reading Folland's A Course in Abstract Harmonic Analysis and find this book extremely exciting. However it seems Folland does not give many examples to illustrate the motivation behind much of ...
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1answer
34 views

reduced density matrix for the given composite system

Given the composite system of two qubits $$ |\psi^{AB}\rangle=\frac{1}{\sqrt{2}}(|0^{A}\rangle \otimes|0^{B}\rangle+|1^{A}\rangle\otimes|1^{B}\rangle) $$ with the density matrix of the composite ...
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votes
1answer
30 views

Double dual of the space of bounded operators on Hilbert space [duplicate]

Every Banach space $X$ is canonically, isometrically embedded in its bidual $X^{**}$. But it is not always $1$-complemented in the bidual: for example, there is no projection from $\ell_\infty$ onto ...
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1answer
24 views

Weak operator topology convergence of hermitian operators

Let $\{A_i\}$ be a net of hermitian operators on a separable Hibert space $\mathbb{H}$ and suppose that there is a hermitian operator T such that $A_{i}\le T$ for all i. If $\{<A_i h,h>\}$ is an ...
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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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Homotopy of bounded homomorphisms between Banach algebras

Let $A$ and $B$ be Banach algebras. Say that two bounded homomorphisms $\phi_0$ and $\phi_1$ from $A$ to $B$ are homotopic if there is a path $(\phi_t)_{t\in[0,1]}$ of bounded homomorphisms from $A$ ...
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1answer
24 views

Second countable property for abelian von neumann algebras

I am looking at Murphy's book "$C^*$algebras and operator theory", in the section on abelian Von Neumann algebras (end of chapter 4). There, it is explained that any (unit containing) abelian Von ...
5
votes
2answers
415 views

On the exponential form of a unitary matrix

A unitary matrix $U \in \large C_{n,n}$ can always be written in the exponential form: $U = e ^{iA}$ (1) where $A$ is Hermitian. My goal is to find the Hermitian matrix $A$, given the unitary ...
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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 ‎‎$‎‎K(H)‎\subseteq ‎\varphi‎(A)$ ...
2
votes
0answers
24 views

Path of completely bounded maps has uniformly bounded cb norm?

If $\phi_t:A\rightarrow B$ is completely bounded for $t\in[0,1]$, and $t\mapsto\phi_t(a)$ is continuous for each $a\in A$, is $\sup_{t\in[0,1]}||\phi_t||_{cb}$ finite? Here, $A$ and $B$ are operator ...
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vote
2answers
55 views

Classification of Banach Algebras?

Is there a classification theorem for Banach algebras, or even for Banach *algebras, similar to the GNS representation theorem for $C^*$-algebras? If yes, please provide a reference where I can read ...
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votes
3answers
987 views

Applications of Banach Algebras and Operator Algebras

I am trying to learn operator algebra theory (I am tempted to start with Douglas' "Banach Algebra Techniques in Operator Theory"). One aspect that I am curious about is whether there are significant ...
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0answers
33 views

Polar decomposition for completely bounded linear maps

Let $M$ be a W*-algebra and $f$ be a norm one and normal functional on $M$. Polar decomposition says that, there is a unique positive linear functional, denoted by $|f|$, satisfying in: ...
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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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1answer
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Understanding minimal projections

This might be very easy but it is not quite clear for me. Detailed explanation appreciated! I went through the commutative case but beyond that I lack intuition. Let $A$ be a C*-algebra and let ...
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1answer
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which von Neumann algebras have many sufficiently normal irreducible representations?

Let $M$ be a von Neumann algebra. We say that $M$ has many sufficiently normal irreducible representations, namely $\{\pi_i\}$, if $||a||=\sup ||\pi_i(s)||$. For example the second dual of a ...
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1answer
27 views

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
81 views

Theorem about irreducible representation of $C^*$-algebra

I have been told, that there is a theorem about irreducible representation of $C^*$-algebras, but I have troubles finding it. It is also possible, that this theorem is consequence of some theorem I've ...
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votes
1answer
30 views

*-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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1answer
34 views

Coherent states - operator algebra problem with physics motivation

Motivation: I have a mathematical problem motivated by quantum field theory in physics. It should be quite easy to prove, but for some reason I can't do it. Intro: Let there be operators $\hat{a_i}$ ...
5
votes
1answer
173 views

Minimal projections vs maximal left ideals

I've seen in some papers a statement (which is referred to a very old book of Dixmier in French which I have no access to / can't read anyway) saying that maximal left ideals of a (unital) C*-algebra ...
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votes
1answer
30 views

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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vote
1answer
58 views

If an operator have only Real eigenvalues + symmetric then it's self-adjoint?

I know that if an operator is self-adjoint then has Real eigenvalues but I'm not sure about the converse i.e. if it has only Real eigenvalues and is symmetric then the operator is selfadjoint. Is that ...
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votes
0answers
26 views

In a proof of the theorem about the abstract index group of a Banach algebra

The following is a proposition in the Banach Algebra Techniques in Operator Theory by Douglas: I don't quite understand the very last step of the proof. Let $\pi:G\to G/G_0$ be the cannonical ...
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0answers
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 ...