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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Commutant of algebra of multiplication operators

Let $L^2(X)$ be the set of Lebesgue square-integrable functions on a locally compact Hausdorff space $X$. Define $\mathfrak{A}:=\{M_f:f\in L^{\infty}(X), f=\overline{f}\}$, where $M_f$ is the the ...
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Applying holomorphic functional calculus to produce idempotents in a Banach algebra

Let $a$ be an element of a unital Banach algebra $A$. Suppose the spectrum of $a$ is contained in $B(0,\sqrt{\varepsilon})\cup B(1,\sqrt{\varepsilon})$ where $B(z,r)$ denotes the open ball of radius ...
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almost unital Banach algebra's

Let $A$ be an "almost unital Banach algebra", in the sense that it satisfies all the usual axioms but not necessarily that $\|1\|=1$. From the product inequality $\forall x,y \in A$ \begin{equation} ...
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Existence of central cover for a representation of a C*-algebra

I've been trying to learn the basics about the representation theory of C*-algebras and came across the following in Pedersen's C*-algebras and their Automorphism Groups: With each ...
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Is $\|a^2\|≤\|\Gamma(a)\|\|a\|$right?

For any $a$ in a commutative Banach Algebra with Gelfand representation $\Gamma$, is this inequality $\|a^2\|≤\|\Gamma(a)\|*\|a\|$right? If so, how to prove it? I'm exhausted for many hours' attempt ...
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understanding a theorem in c*algebra

I want to understand the following theorem: Theorem: Let A and B $C^*$-algebras with A unital, and let $\varphi:A\to B$ a bounded linear selfadjoint map such that: for every self-adjoint elements ...
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Existence of certain subalgebras of $C(X)$

Suppose $A$ is a commutative Banach algebra.Knowing that the Gelfand transform is not surjective but injective does it imply that $A$ is not isomorphic to $C(X)$ ? by $M$ we mean the maximal ideal ...
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Characterisation of a Commutative C* Algebra which is an Integral Domain

Let $X$ be a compact hausdorff topological space with more than one element.Then prove that the ring $C(X)$ of complex valued continuous functions on $X$ is not an integral domain. Thanks for any ...
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Interpreting the lingo of a definition

The Terms I grew up with: A bounded linear operator $U$ on a Hilbert space $H$ is a partial isometry if there exists a subspace $M$ of $H$ such that $\|Ux\| = \|x\|$ for all $x\in M$, and $Ux = ...
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26 views

Involution on the Disc Algebra

We know that the disc algebra $A(D)$={$f$ in $C(DUBd(D))$: f is analytic on $D$} wher $D$ is the open unit disc in the complex plane is a Banach algebra under the usual sup norm.Is it possible to ...
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51 views

Counterexample for an isometric homomorphism of algebras which is not involutive.

I am finding difficulties in finding a counterexample that if $f:A\to B$ is a homomorphism of $C^*$algebras A and B (which means: f is linear and multiplicative) and let f be isometric, this implies ...
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Herz-Schur multiplier bounded if corresponding functional is bounded

I want to prove the following statement: Let $\Gamma$ be a discrete group and $\phi:\Gamma\rightarrow\mathbb{C}$ a function and ...
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30 views

Tensor product of $C^*$- algebras

We know from the paper of Douglas and Howe (enter link description here) that the commutator ideal $\mathcal{I}$ of $\mathcal{A}(C(T^2))$, the $C^*$-algebra generated by Toeplitz operators with ...
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Order zero maps in matrix algebra

Let $a$ and $b$ are two elements in a $C^*$algebra $A$. We say $a\perp b$ if $ab=ba=a^*b=ab^*=0$. We say a completely positive map $\phi: A \rightarrow B$ is of order zero if for any positive elements ...
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Where does the double commutant theorem fails for $AW^*$-algebras?

Commutative $AW^*$-algebra are characterized as those $C^*$-algebras such that their space of projections is a complete boolean algebra (see http://en.wikipedia.org/wiki/AW*-algebra). Von Neumann ...
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Existence of idempotents versus existence of projections in a C*-algebra

Let $\mathcal{A}$ be any C*-algebra. Suppose $x\in\mathcal{A}$ is idempotent, with $x\neq 0$ and $x\neq 1$. Does it follow that $\mathcal{A}$ admits nontrivial projections? Clearly, when $x$ is ...
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Non-closed ideals in $C^*$-algebras

What is an example of an ideal in a commutative $C^*$-algebra that is not closed? If by chance every ideal in a commutative $C^*$-algebra is closed, how about is non-commutative $C^*$-algebras? ...
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34 views

Norms on unitization of nonunital Banach algebra

Let $A$ be a nonunital Banach algebra and denote by $A^+$ the unitization of $A$. One commonly used Banach algebra norm on $A^+$ is given by $||(a,\lambda)||=||a||+|\lambda|$ (where $a\in ...
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C^* algebra generated by a C^* algebra and a group

In this article: http://ac.els-cdn.com/S0022123606001856/1-s2.0-S0022123606001856-main.pdf?_tid=ddbd529e-b857-11e4-9f30-00000aab0f27&acdnat=1424365005_7c9a5ac14284f6258613e8a6cb8a9482 "Spectral ...
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44 views

when a crossed product group is inner amenable

Denote $K, H$ to be countable discrete groups, then I am interested whether the crossed product group $G=H\rtimes_{\alpha} K$ is inner amenable or not. For example, when $\alpha$ is trivial, ...
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Hilbert space structure on $C^{*}$ algebras

What is an example of an infinite dimensional $C^{*}$ algebra with a Hilbert space structure (not merely pre-hilbert structure) such that the orthogonal complement of each closed left ideal ...
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Are ideals generated by separable subspaces separable?

Suppose that $X$ is a compact Hausdorff space and take a sequence $(f_n)$ in $C(X)$ such that the ideal generated by $(f_n)$ is proper. Must this ideal be separable as a Banach space? It looks to me ...
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C* Algebra Positivity

STATEMENT: This is a proof from one of Qiaochu's notes on $C^*$ algebras. Proof: Let A be a $C^*$ algebra.We now want to show that for any $c\in A$ we have $c^*c\geq 0$. Suppose otherwise.We know ...
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Infinite dimensional C*-algebra contains infinite dimensional commutitive subalgebra

I was reading a paper which mentioned without proof that every infinite-dimensional $C$* algebra has an infinite-dimensional commutative $C$* subalgebra. Thinking about it for 10 minutes, I didn't ...
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Spatial tensor product of operator spaces

If $X$ and $Y$ are Banach spaces and $\otimes_\varepsilon$ denotes the injective tensor product, then in general $\otimes_\varepsilon$ does not respect quotients unless we map into ...
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A question about induced $C^\ast$-algebra

Recently, I read the book Crossed Products of C*-algebras, and meat a question. The question is how to prove $\mathrm{Ind}_c(A,\alpha)$ is dence in $\mathrm{Ind}(A,\alpha)$. On the page 102, the ...
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Group $C^*$-algebra elements as limit of self-adjoint integrable functions

Assume $G$ is a locally compact abelian group and let $C^*(G)$ denote its group $C^*$-algebra. I am reading a proof that uses the 'fact' that some $f\in C^*(G)$ is a limit of self-adjoint functions ...
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49 views

Partial Isometries: Characterization

Given a C*-algebra. Any partial isometry satisfies: $$WW^*W=W$$ From this, one derives projections: $$W^*W,WW^*$$ Conversely, given projections: $$W^*W,WW^*$$ One derives a partial isometry: ...
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Extending isomorphisms between $*$-algebras to $C^*$-algebras

I'm quite sure I am correct about this but at the moment I can't think for the life of me why. Suppose $A$ and $B$ are $*$-algebras and there are $*$-homomorphisms $\pi_1 \colon A \to ...
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A symmetric algebra that is not a C* algebra

Recall that a commutative Banach $*$-algebra $A$ is called symmetric if the Gelfand transform replaces involution in $A$ by complex conjugation in $\mathbb{C}$. Moreover, any commutative C* algebra is ...
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Dynamics: Continuity

Disclaimer: This is a record of results. Given a C*-algebra $\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$. Consider a Hamiltonian dynamics: ...
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States: KMS-Condition

Given a C*-algebra $\mathcal{A}$ with dynamics $\tau$. Consider a state $\omega$. Does it suffice to have on a dense set the KMS-condition: ...
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Commutators of Schur polynomials of Lie algebra elements

Question: Is there a well-known formula for computing the commutators of Schur polynomials when the variables are Lie algebra elements? If the algebra has a particularly simple commutation relation, ...
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225 views

Homotopy classes of $*$-morphisms and unital $*$-morphisms

Let $A$ and $B$ be C*-algebras (non necessarily unital). A homotopy between two $*$-morphisms $\phi,\psi:A \to B$ is a $*$-morphism $A \to C([0,1],B)$ such that you can recover $\phi$ and $\psi$ from ...
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Tensor Products of C*-Algebras

If A, B are C*-algebras, show that there exists a unique $*-isomorphism $ $ ‎\theta‎‎: A ‎\otimes‎_{*}‎‎ B ‎\longrightarrow‎ B ‎\otimes‎_{*}‎‎ A $ such that $\theta( a \otimes‎_{*} b) = b ...
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Question about $C_0(X)$-algebras and $C_b(X)$.

Let $X$ be a locally compact Hausdorff space. Denote by $C_0(X)$ its C*-algebra of continuous functions that vanish on infinity and by $C_b(X)$ its C*-algebra of bounded functions. Now, let $A$ be a ...
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Spectral decomposition of a an Operator on $\mathcal{L}(l^2(\mathbb{N})$

I have an exercise, where I do not really know how to solve it. Let $\{\lambda_n\}_n$ be a counting of the rational numbers in $[0,1]$ and define a bounded self-adjoint operator $T$ on ...
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Self adjoint operator has spectrum around 0 and 1

I need to prove the following statement: Let $\mathcal{A}$ be a unital $C^*-$Algebra, $A$ a self-adjoint element and $P$ a projection, so $P^2=P=P^*$. Let $\delta :=\|P-A\|$. I want to prove that ...
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Power of positive operator

Let $H$ be a complex Hilbert space and $B(H)$ be the space of all bounded linear operator on $H$. Let $T\in B(H)$ be a positive operator ($\langle Tx,x\rangle\geq0$ for all $x\in H$) and $\alpha\in ...
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States: Liouvilleans

Given a C*-algebra $\mathcal{A}$ with dynamics $\tau$. Consider an invariant state: $\omega\circ\tau^t\equiv\omega$ Then the dynamics is unitarily implementable: ...
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Proving the unitary relation of ensemble decompositions

I apologize if this is better suited for physics.stackexchange; looking at previously asked questions it seems as if this would be a good fit here, as the arguments are mostly mathematical. In my ...
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States: Approximate Unit

Given a C*-algebra $\mathcal{A}$. Consider a state: $\omega\geq0$ Especially one has: $\sup\omega(E^2)=\|\omega\|$ Can it actually fail to be a proper limit? The problem is that the square is not ...
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Positive Elements: Norm (Decomposition)

Given a C*-algebra $\mathcal{A}$. Then every element decomposes into: $Z=X_+-X_-+iY_+-iY_-=\sum_{\alpha=0\ldots3}i^\alpha Z_\alpha$ Obviously, one has: $\|Z\|\leq\sum_{\alpha=0\ldots3}\|Z_\alpha\|$ ...
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States: Positivity vs. Continuity

Definition Define positivity in terms of: $$\omega\geq0:\iff\omega(X^*X)\geq0$$ (This way it serves best for the GNS-construction.) Problem Given a C*-algebra $1\notin\mathcal{A}$. Consider a ...
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States: Extension

Definition Define positivity in terms of: $$\omega\geq0:\iff\omega(X^*X)\geq0$$ (This way it serves best for the GNS-construction.) Problem Given a C*-algebra $1\notin\mathcal{A}$ and adjoin a unit ...
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An elementary perturbation result in C*-algebra

The following question was raised when I read a papaer "MF actions and K-theoretic dynamics". In one of the proofs in that paper, the author utilized a so called "elementary perturbation result in ...
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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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Approximate Identity: Projections

Problem Given a C*-algebra $\mathcal{A}$. Denote the positive open unit ball by: $\mathcal{B}^+$ Then it has an approximate identity: ...
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Positive Elements: Decomposition?

Problem Given a C*-algebra with unit $1\in\mathcal{A}$. Then every selfadjoint element decomposes into positives ones: $$A=A^*:\quad A=\frac12(|A|-A)-\frac12(|A|-A)\quad\left(\frac12(|A|\pm ...
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Dynamics: EQ-States vs. NESS-States

Given a C*-algebra $\mathcal{A}$ with dynamics $\tau$. Consider a state that relaxes towards equilibrium: $$\omega_T(A):=\omega\circ\tau^T(A)\stackrel{T\to\infty}{\to}\omega_\infty(A)$$ Then it ...