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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Group von Neumann algebras

I have a question about group von Neumann algebras structure. If $L(G)$ is a subset of $L(H)$, can we find a subgroup $G_1$ of $H$ such that $L(G_1)$ is isomorphic to $L(G)$? I appreciate any help.
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Continuous family of subalgebras in a C* algebra

Let A be a separable C* algebra. Fix $n\in\mathbb N$. For t $\in$ $\mathbb{R}$ let $A_t$ be a subalgebra of $A$ such that: $A_t \cong \mathcal{O}_n$ (Cuntz algebra). Generators of $A_t$ depend ...
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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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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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386 views

application of c*algebras to PDEs

I am preparing an introductory talk about c* algebras and I'd like to motivate C*algebras or show an application of them and I'd prefer an application in the field of partial differential equations. ...
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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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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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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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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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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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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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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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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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The inverse limit of C$^*$-algebras and whether it commutes with taking the minimal tensor product

Suppose we are given a C$^*$-algebra $A$ and a family of C$^*$-ideals $\mathfrak{I}$ that is upwards directed when ordered by reverse inclusion (i.e. for any $I_1,I_2\in\mathfrak{I}$ there exists a $J\...
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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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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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$T^p$ 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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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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Commutator formula in infinite dimensions

The commutator formula states that for $A,B$ elements of a Lie algebra, $$ \lim_{n\to \infty}\left\{ \exp\left(-A\tfrac{t}{n}\right)\exp\left(-B\tfrac{t}{n}\right)\exp\left(A\tfrac{t}{n}\right)\exp\...
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108 views

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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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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QD C*-algebra's representation theorem

Here is a question from the proof of the "QD C*-algebra's representation theorem" in P245 of book "C*-algebras and Finite-Dimensional Approximations" by Nate and Taka. For a separable unital C*-...
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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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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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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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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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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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Operator - Exponential form

It is well known that for every unitary operator $\hat U$ an exponential of the form $$ \hat U = e^{i\hat H} $$ exists ($\hat H$ is hermitian). But I can only prove it the other way round: $$ (e^{i\...
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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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Tensor product of algebras of operators on $SQ_p$ spaces

I am aware that there is a canonical tensor product for $L_p$ spaces that allows one to talk about tensor products of algebras of bounded linear operators on $L_p$ spaces. Is there an analogous notion ...
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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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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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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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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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A certain limit of a state is zero

Let A be a C$^*$-algebra, $a\in A$ strictly positive (this means: for every state $\varphi$ of A is $\varphi(a)>0$). Let $u_n=(\frac{1}{n}+a)^{-1}$. Then for all $b\in A$ and all states $\varphi$ ...
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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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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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If $\|p-q\|<{1\over2}$ then $p$ is homotopy equivalent to $q$

Let $A$ be a $C^*$ algebra, $p,q \in A$ projections, such that $\|p-q\|< {1 \over 2}$. Show that $p$ homotopy equivalent to $q$. Proof. Let $a_t=(1-t)p+tq$, then $a_t$ is positive (self-adjoint ...
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Nontrivial closed ideal of $\mathbb{B(H)}$, $\mathbb{H}$ is a non-separable Hilbert space.

$\mathbb{H}$ is a non-separable Hilbert space. Give an example of nontrivial closed ideal $I$of $\mathbb{B(H)}$, that is different from $\mathbb{B_0(H)}$ which is the ideal of compact operators. Any ...
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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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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}}=\...