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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A nilpotent element of an algebra which does not lie in the span of commutator elements.

What is an example of a $C^{*}$ algebra such that the span of nilpotent elements is not a sub vector space of the span of commutator elements. Obviously any such $C^{*}$ algebra would be a ...
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1answer
21 views

Does pointwise nilpotency imply global nilpotency?

Is there a compact Haussdorf space $X$ and $C^{*}$ algebra $A$ with a continuous map $f:X\to A$, such that $f(x)\in A$ is a nilpotent element, $\forall x \in X$, but $f$ is not a ...
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24 views

Equivalence of Definitions of Strong Operator Topology

I have a couple questions about how we define the strong operator topology on $\mathscr{B} (H)$ that I'm hoping someone can help me with. First, I thought that the strong operator topology was the ...
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1answer
45 views

Homotopy elements in C*-algebras

Let A is a C*-algebra, A$^+$ means A$\times$$\mathbb{C}$ equipped with pointwise sum and with a multiplication defined by: (a, $\lambda$)(b, $\mu$) = (ab + $\lambda$b + $\mu$a, $\lambda$$\mu$) If x, ...
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35 views

Atomic W*- algebra

Let $A$ be a C*-algebra. Put $z: = \sup\{ e\in A^{**} ;\text{ e is a minimal projection}\}$. Easily can see $z$ is a central projection. Set $M:= A^{**}z$. 1) Is $M$ an atomic W*-algebra in general? ...
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1answer
67 views

$C(S^1)$ does not have a single generator

Let $S^1$ be the unit circle in the complex plain and $C(S^1)$ be the continuous function space on $S^1$.$f\in C(S^1)$ is a generator means that $\{p(f) |\text{ p is a polynomial in z}\}$ is dense in ...
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1answer
42 views

Direct limits of simple C*-algebras are simple

Let $S$ be a non-empty set of simple C$^*$-subbalgebras of a C$^*$-algebra $A$. Let us also suppose that $S$ is upwards-directed and that the union of all element of $S$ is dense in $A$. Then $A$ is ...
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1answer
39 views

How is the spectrum of a product of operators related to the spectrum of each term in the product?

I will use the usual notation of $\sigma(A)$ to denote the spectrum of an operator $A$. Is there a relationship between the spectrum of bounded operators (on complex Hilbert space) $A$ and $B$ and ...
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91 views

Comparing two positive linear functionals when support$(\psi)\leq$ support$(\phi)$

The following point has been addressed in the significant paper "order ideal in C*-algebras and its dual (By E. Effros - Lemma 4.1). The reference is Dixmier's book (1957 French). Unfortunately, I ...
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2answers
48 views

Functional calculus for unitization of an algebra?

I have been stuck on this problem for a week now: "Let $A$ be a Banach algebra without identity, let $a\in A$, and let $f$ be holomorphic on a neighborhood of $\sigma(a)$, so that $f(a)\in A^\# $ is ...
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1answer
54 views

$C*$-algebra Identity for Matrices

What's the easiest way to see that the $n \times n$-matrices over $\mathbb{C}$ satisfy the $C^*$-algebra identity $\|aa^*\| = \|a\|^2$?
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1answer
30 views

One-sided identities in Banach algebras

What is an example of a Banach algebra with a left identity but with no right identities? Is there an example of an operator algebra with this property?
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20 views

Special case of Green's imprimitivity theorem and related question

Consider a locally compact group $G$ and a closed subgroup $H$ of $G$, and let $G$ act on $G/H$ by left translation. Green's imprimitivity theorem implies that the crossed product $C_0(G/H)\rtimes G$ ...
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0answers
65 views

The reduction of nilpotency order of nilpotent elements of $C^{*}$ algebras

Assume that $A$ is a unital $C^{*}$-algebra. Let $a\in A$ be a nilpotent element with $$a^{k}=0,\;\;k>1.$$ Are there two elements $x,y\in A$ with $a=xy,\;\;(yx)^{k-1}=0$? Motivation for ...
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1answer
30 views

Expectation value of the product of two operators

Does $\langle A^2B^2\rangle = \langle B^2A^2\rangle$ (i.e. the operators $A$ and $B$ commute) somehow imply $\langle A^2\rangle = \langle B^2\rangle$? If so, why? To add some context, my operators ...
2
votes
1answer
40 views

A certain *-isomorphism

let $A$ be a C*-algebra and $z\in A^{**}$ the supremum of all the minimal projections in $A^{**}$. How can show $* -$ homomorphism $A\to zA\subset zA^{**}$ is injective?
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18 views

Understanding Operators in context of Green's function derivation

I am trying to understand what operators actually mean when deriving the definition of green's function. What is the interagl representation of an operator? Is this correct? $$ D = <x|\int D|x> ...
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1answer
74 views

Is the algebra of adjointable operators on a Hilbert module prime?

In abstract algebra, a nonzero ring $R$ is a prime ring if for any two elements $a$ and $b$ of R, $arb = 0$ for all $r$ in $R$ implies that either $a = 0$ or $b = 0.$ Or for any two ideals $A$ and $B$...
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28 views

Is there a sample-path continuous stochastic process whose sample paths do not almost surely lie in an RKHS?

Let $f$ be a mean zero second-order stochastic process with continuous covariance function $k$, that is indexed on a separable metric space $\mathcal{X}$ and that is sample-path continuous. Can we ...
6
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1answer
118 views

Can $xy$ and $yx$ lie in different connected components of the group of invertible elements of an algebra?

What is an example of a Banach or $C^{*}$ algebra $A$ which has two invertible elements $x, y$ such that $xy$ can not be connected to $yx$ in $G(A)$, the space of invertible elements of $A$. A ...
2
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1answer
71 views

Can all compact subsets of $\mathbb{C}$ be spectra of a bounded operator on $C[0,1]$?

Let $K$ be a non-empty compact subset of $\mathbb{C}$, the complex field. Does there exist an operator $A\in \mathscr{B}(C[0,1])$ such that $\sigma(A)=K$? $A$ is a multiplication operator iff $K$ is ...
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21 views

Binormal operator - equivalent definitions?

I have seen two different definitions of a binormal operator A. A is unitarily equivalent to a block 2x2 matrix of commuting normal matrices. AA* commutes with A*A. I am hoping these definitions ...
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2answers
115 views

How does the general spectral theorem generalize the simpler versions?

I read the following version of the spectral theorem in Banach Algebra Techniques in Operator Theory by Douglas: I'm trying to understand why this is a generalization of the following version, ...
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1answer
60 views

The algebras of compact operators on $\ell_p$ as direct limits of matrix algebras

Consider $M_n(\mathbb{C})$ as $B(\ell_p^n)$ for $n\in\mathbb{N}$ where $p\in[1,\infty)$, and include $M_n(\mathbb{C})$ in $M_{n+1}(\mathbb{C})$ as the upper left corner. Is it true that $\overline{\...
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1answer
69 views

Does $C^*(G) \cong C^*(H)$ imply that $\mathbb{C}G \cong \mathbb{C}H$?

I wonder whether the underlying complex group algebra of a group $C^*$-algebra is unique? I.e. if $G$ and $H$ are discrete groups such that $C^*(G) \cong C^*(H)$ (or $C^*_r(G) \cong C^*_r(H)$) as $C^*$...
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60 views

Difference between Schmidt decomposition and singular value decomposition

Schmidt decomposition of an operator is a useful tool of quantum information theory nowadays. Let $O$ be an operator acting on the Hilbert space $\mathcal{H}_{d_1} \otimes \mathcal{H}_{d_1}$. $\...
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1answer
34 views

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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1answer
57 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 \, ...
2
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1answer
31 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 $\tau \leq 1$ $e_ie_j=...
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40 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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24 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 $S$...
2
votes
1answer
46 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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1answer
31 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 $\{y_1,...,y_n\}\...
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0answers
28 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 ...
4
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1answer
64 views

Idempotents which are not Murray-von Neumann equivalent to its adjoint

What is an example of a $C^{*}$ algebra with an idempotent $e$ such that $e$ is not Murray-von Neumann equivalent to $e^{*}$?
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1answer
25 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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27 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 C(...
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0answers
12 views

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
27 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 references?...
3
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55 views

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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11 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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30 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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21 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 $I\cup\mathscr{B}...
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1answer
41 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 $\...
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0answers
24 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 $(a,z)(b,w)=(ab+zb+wa,...
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45 views

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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1answer
27 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 $I_{\alpha}$...
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1answer
52 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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29 views

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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1answer
29 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 ...