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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Involution and Gelfand Transform Properties

Let $\mathcal{B}$ be a commutative unital Banach algebra, and let for each $x\in\mathcal{B}$ $\hat{x}$ be the Gelfand transform. I assume that $\mathcal{B}$ has an involution *. I want to show that: ...
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The set of all maximal ideals (Wiener Algebra)

I am trying to prove a proposition and in my proof I somehow need to find the set of all maximal ideals of a Banach Algebra. This is my working environment: Let $A(\mathbb{R}^2)$ be the (Wiener ...
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Inclusions of Multiplier algebras associated to hereditary subalgebras

I have been searching for a proof of the following fact. Let $A$ and $B$ be C$^\ast$-algebras such that $A$ is a subalgebra of $B$ (in the C$^\ast$-algebraic sense of course) and Let $C = \overline{...
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noncommutative algebra

If we let $B$ be a noncommutative Banach algebra with unit $e$, then, obviously, $xy\not=yx$, but are they related? I suspect that the spectral radii of $xy$ and $yx$ are the same, but I couldn't ...
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Algebra $A$ and its Gelfand spectrum

Let $A$ be the set of all function $f$ on $\mathbb{R}$ of the form $$ f(x)=d+\int\limits_{0}^{\infty}e^{ixt}k(t)dt,\qquad\quad x\in\mathbb{R}, $$ where $d\in\mathbb{C}$ and $k\in L_1([0,\infty])$. The ...
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Connected components of group of unitaries in Calkin algebra

Let $H$ be a separable infinite dimensional Hilbert space. Denote the Calkin algebra by $Q(H)=B(H)/K(H)$, and $U(Q(H))$ the group of unitaries in $Q(H)$. I'm trying to show that the map $F: U(Q(H))/...
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Commutative Banach algebra and its Gelfand spectrum

Let $A$ be the set of all functions on $\mathbb{R}^2$ of the form $$ f(t,s):=\sum_{m=-\infty}^{\infty}\sum_{n=-\infty}^{\infty}{a_{mn}e^{i(mt+ns)}}, $$ with the following norm: $$ \|f\|:=\sum_{m=-\...
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Commutative Banach algebra and its maximal ideal space

Let $A:=C^{(n)}([0,1])$ be the set consisting of the n-times continuously differentiable complex-valued functions. Consider $A$ with the norm $$ \|f\|:=\max\limits_{0 \leq t \leq 1} \sum_{k=0}^{n}{\...
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Definition of second topological $K$-group of a Banach algebra

The question is a about the definition of the second topological $K$-group of a Banach algebra $A$. I was reading a text of Alain Valette (Prop. 3.3.7) where he proves that $$ K_1(SA) \cong \pi_1(\...
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Unitary element in an AF $C^*$-algebra can be approximated by sequence of unitaries

Let $A$ be a unital AF $C^*$-algebra. Write $A=\overline{\bigcup_{k\in \mathbb{N}}A_k}$ where each $A_k$ is a unital (with the same unit of $A$) finite dimensional $C^*$ subalgebra. Suppose $u\in A$ ...
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Over sequential spaces and $B(H)$

We say that a topological space $X$ is sequential if the following holds : If $U$ is sequentially open then $U$ is open. By sequentially open we mean that $x \in U$ and $x_n \to x$ implies that $x_n$ ...
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continuous and sequentially continuous

If an operator $T: A\rightarrow B$ satisfying for every sequence $\{X_n\}$ weakly converging to $X$, we have $TX_n \rightarrow TX$ in weak topology. Then, is $T$ weak-weak continuous? And in the WOT/...
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There is no trace on Cuntz algebra

Here is a general explanation why purely infinite $C^*$-algebras admit no tracial states: Non-existence Tracial states. Is my following explanation for non existence of trace on Cuntz algebra $O_n$ (...
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SOT-isomorphic C*-algebras

Suppose that $A, B \subset B(\mathcal{H})$ are $C^*$-algebras. Assume that $\{p_n\} \subset B(\mathcal{H})$ is a monotone sequence of projections such that: $p_n \rightarrow 1$ in strong operator ...
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Projective limit of finite dimensional C* algebras

Let $A$ be a separable unital $C^*$-algebra and $A$ = $I_0 \supset I_1 \supset I_2 \supset \ldots$ Be a sequence of ideals in $A$ such that: $I_k$ is ideal in $I_m$ when $k \geq m$ $\bigcap I_k = \...
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21 views

$T^p$ increases to $T$ in strongly operator topology or not.

In a Hilbert space $H$, let $T$ be a positive operator on $H$ with $\|T\|_\infty\le 1$. Then, obviously, $T^p$ is increasing as $p$ decreases to 1. But I am not sure whether $T^p$ increases to $T$ in ...
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40 views

Applications of functional calculus? [closed]

I been trying to figure out functions other then projections that are of intrest in the settings of functional calculus, but I cant. Can anyone help me shed light on this? I.e what problems other ...
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Correspondence between maximal ideals and multiplicative functionals of a non unital, commutative Banach algebra.

Let $\mathcal{A}$ be a non (necessarily) unital commutative Banach algebra, and let $$ M_{\mathcal{A}} = \{ \phi:\mathcal{A} \to \mathbb{C} : \phi \mbox{ is multiplicative and not trivial}\} $$ and $$...
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Lemma V.1.25 (page 298) from the book of Takesaki (Vol 1).

I have a problem with the following lemma (Takesaki Vol 1-page 298): Lemma 1.25. If $e$ is an abelian projection in a von Neumann algebra $\mathcal M$, then for any projection $f\in\mathcal M$ ...
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Universal $C^*$ algebras

It is known that the $C^*$-algebra $\mathcal U$ generated by bilateral shift $\ell^2 (\mathbb Z) \ni e_k \mapsto e_{k+1} \in \ell ^2(\mathbb Z)$, is a universal $C^*$ algebra generated by unitary: for ...
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about representations of a simple $C^*$-algebra

We know that every simple $C^*$-algebra is primitive, say it has a faithful non-zero irreducible representation. The converse is not necessarily true. An counterexample is just the $B(H)$ when $H$ is ...
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Not every positive operator is positive-definite operator

According to the definitions for the operator $L: H \rightarrow H$ we have: $L$ is positive operator if the inner product $\langle Lu\mid u \rangle \geq 0$ for $\forall u \in H$ $L$ is positie-...
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What is the maximal ideal space of $H^\infty$?

What is the spectrum of $H^\infty$, the Banach algebra of all bounded holomorphic functions in the open unit disk $D=\{z\in \mathbb{C}\mid |z| <1 \} $?
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almost unitaries are close to a unitary element

I need help to prove the following exercise: Let $\epsilon >0$. Show that there exists $\delta >0$ with the property: If $A$ is a unital $C^*$-algebra and $x\in A$ such that $\|x^*x-1\|<\...
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Does tensor product with $L_p$ operator algebra preserve exact sequences?

By $L_p$ operator algebra I mean a closed subalgebra of the algebra of bounded linear operators on some $L_p$ space where $p\in(1,\infty)$. There is a notion of tensor product of $L_p$ spaces (as ...
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What is the dual space of a von Neumann algebra?

What is the dual space of a von Neumann algebra $\mathcal{M}$? Does it have any specific form? Or just $\mathcal{M}^*$.
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All nonzero singular values of $A$ are equal to $1$ iff $A^*=A^*AA^*$ and $A=AA^*A$

I want to show that all the non-zero s-numbers, i.e. singular values $s_j(A):=(\lambda_j(A^*A))^{1/2}$, of A (a bounded linear operator of finite rank acting on a separable Hilbert space $H$) are ...
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Equivalence of some characterization of pure states

I'm looking for a reference or a proof for these well-known facts in $C^*$-algebras theory for which, however, I havent found any clearly written proof of the same type of ones I will sketch. ...
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Uncountable sequence $R_1 \subset R_2 \subset … $ of von Neumann algebras acting on separable $H$?

Can there exist an uncountable sequence $R_1 \subset R_2 ...$ of von Neumann algebras all acting on the same separable Hilbert space $H$, with a "limit" algebra $R$ such that $R_\alpha \subset R$ for ...
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Show that $A\varphi_j=\left<A\varphi_j,\varphi_j\right>\varphi_j$ and $A^*A\varphi_j=s_j(A)^2\varphi_j$ for all $j$

Let $A$ be a bounded linear (compact) operator acting on a separable Hilbert space $H$, and let $\varphi_1,\varphi_2,\ldots$ be an orthonormal basis of $H$. I Assume that $|\left< A\varphi_j,\...
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Semi-finite trace on a von Neumann algebra: Equivalent definitions

Let $(N,\tau)$ be a semi-finite von Neumann algebra. This means that $\tau$ is a normal, faithful and semi-finite trace. Normality means that $\tau(x) = \sup_i \tau(x_i)$ if $x \in N_+$ is the limit ...
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modular operator

I could calculate a modular operator $Δ$ for a state $τ_α(x_{ij})= αx_{11}+(1-α)x_{22}(0<α<1)$. But, I cannot understand an automorphism $σ_t(x)=Δ^{it}(x)Δ^{-it}$ of $M_2(\mathbb{C})$. For ...
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1answer
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Compute the positive part of $K_0(A)$ where $A$ is a simple AF algebra

I'm trying to understand the following example from my lecture notes: Define $A_n=M_{F_n}(\Bbb{C})\oplus M_{F_{n+1}}(\Bbb{C})$ where $F_n$ defined by $F_1=1, \ F_2=2, \ F_{n+2}=F_{n}+F_{n+1}$, i.e., ...
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Countable weighted shift has no invariant subspace.

Suppose I have $T(e_n)=w_ne_{n+1}$ where $w_n>0$ (and are bounded) and $\{e_n\}$ denotes the canonical basis of $l^{2}(\mathbb{N})$. I would like to prove that the only (closed) invariant ...
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A question about orthogonality

Let $\mathcal{A}$ be a unital $*$-algebra over $\mathbb{C}$ and let $a,b\in\mathcal{A}$ be projections, that is, $a=a^*=a^2$ and $b=b^*=b^2$. If $a+b=1$, then $ab=0$. This follows from - \begin{align*...
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Continuous family of subalgebras in a C* algebra

Let A be a separable C* algebra. For t $\in$ $\mathbb{R}$ let $A_t$ be a subalgebra of $A$ such that: $A_t \cong \mathcal{O}_n$ (Cuntz algebra for fixed n). Generators of $A_t$ depend continuously ...
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Nonunital C*-Algebra: Proper Ideals

Given a C*-algebra without unit. Does there exist a nontrivial proper ideal that does not lie in a maximally nontrivial proper ideal? (For the unital case this follows easily by Zorn's lemma.) ...
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Bott projection as $K_1$ class

Consider the Bott projection (described in Exercise 5.I of Wegge-Olsen's book $K$-theory and $C^*$-algebras) given by $b(z)=\frac{1}{1+|z|^2}\begin{pmatrix} 1 & \bar{z} \\ z & |z|^2 \end{...
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A comparison between absolute values of functionals

Let $A_0$ be a C*-subalgebra in a C*-algebra $A$. Let $\phi_0$ be a bounded linear functional on $A_0$ and assume $\phi$ is an extension of $\phi_0$ on $A$. I mean $\phi\in A^*$ with $\phi_{|_{A_0}}=\...
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In a C*-algebra, pure states which share the same kernel are equal

I'm reading C*-Algebras by Jacques Dixmier. And in the proof of 2.9.5, it says Let $A$ be a C*-algebra. If $f$ and $f'$ are two pure states which have the same kernel, then $f=f'$. It should ...
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Proving two stubborn inequalities for completely positive maps in C*-algebras

I recently came across this in my studies of functional analysis in C* algebras which got me stuck: For a completely positive map between C* algebras $ \phi : A \to B $ we are to prove these two ...
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Non-self Adjoint Operator Algebra References

The problem I am working on has led me to define a norm closed sub-algebra $\mathscr{A}$ of $\mathscr{B}(\mathscr{H})$. The algebra is generated by some mild relations, and in general, will not be ...
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Spectrum of difference of two projections

Let $p$ and $q$ be two projections in a $C^*$-algebra. What can one say about the spectrum of $p-q$, i.e. is $\sigma(p-q) \subset [-1,1]$ ? The exercise is to show that $\lVert p-q \rVert \leq 1$. ...
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$A\in B(H)$ a unital abelian $C^*$-algebra with cyclic vector then $A'$ is abelian as well

Let $A$ be a unital abelian $C^*$-subalgebra of $B(H)$ (with the same unit as that of $B(H)$), and assume there exists a vector $\xi \in H$ which is cyclic for $A$ (that is, $\{a\xi | a\in A \}$ ...
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Canonical action of a countable discrete group $G$ on its stone-cech compactification $\beta G$

When I read some materials in topological dynamics, I met words: "canonical action of a countable discrete group $G$ on its stone-cech compactification $\beta G$" without any definition. I know that $...
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Elementary proof that $a - 1$ is not invertible, for self-adjoint $a$ with $\lVert a \rVert = 1$

Assume $a \in A$ where $A$ is a unital $C^*$-algebra. If $\lVert a \rVert = 1$ and $a^*=a$ we know that $1 \in \sigma(a)$, the spectrum of $a$. This follows from the fact that $\lVert a \rVert = r(a) =...
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Generalized Absolute Value II

Let $x$ be an operator in $B(H)$. We say a pair $(c,y)$ forms a polar decomposition for $x$ if $y$ is a positive operator, $c$ in $B(H)$ with $x=cy$ such that the restriction of $c$ on $\overline{yH}$...
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Do all $x$ in Hilbert space $H$ equal $U T \eta$ for some $T \in R$ and unitary $U \in R'$, when $\eta$ is separating for $R$?

Let $H$ be a separable Hilbert space on which von Neumann algebra $R$ acts; let $\eta \in H$ be a separating vector for $R$ (i.e. the zero operator is the only $T \in R$ such that $T \eta = 0$); let $...
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Generalized polar decomposition

Let $x\in B(H)$. We say $(x,v,y)$ is a polar decomposition for $x$ if, $\bullet$ $y$ is positive. $\bullet$ $v$ is a partial isometry with $x=vy$. $\bullet$ Ker$(x)$=Ker$(y)$=Ker($v$) The polar ...
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the standard even grading on $M_2(A)$ and $A\otimes \mathbb{K}$

I have a question about a passage in Blackadar's book about K-Theory. Let $A$ be a (ungraded) $C^*$-algebra. There is a grading on $M_2(A)$ with $M_2(A)^{(0)}$ the diagonal matrices and $M_2(A)^{(1)}$...