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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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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Weak convergence of bounded operators

so let $X$ be a Banach space then we say that $A_n \in L(X)$ converges weakly to $A \in L(X)$ if for all $y \in L(X)^*: y(A_n) \rightarrow y(A).$ On the other hand, I just read that weak convergence ...
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Proving an isometric dilation of a non unitary operator on Hilbert space implies infinite dimensional space involving matrices

I have been given this exercise in my Operator theory class dealing with operators on Hilbert spaces, which reads as follows: Let H be a Hilbert space. We are to prove, in two distinct ways, that ...
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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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Integral with Dirac Delta

I've to compute this expression $$ \hat{H} = \frac{1}{4}g_2\int d^3R\int d^3r\ \bar{\Psi}(\vec{R}+\frac{\vec{r}}{2})\bar{\Psi}(\vec{R}-\frac{\vec{r}}{2})\left[ \delta(\vec{r})\nabla_{\vec{r}}^2 +\...
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*-representations of dense subalgebras

Let $H$ be a separable Hilbert space and let $K(H)$ be the C*-algebra of compact operators on $H$. Suppose that $A$ is a *-subalgebra of $K(H)$ which contains all the finite-rank operators. Given a *-...
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continuous depence of the spectrum on elements

Suppose $a_n \to a$ in a unital $C^*$-algebra $A$. If $\lambda_n \in \sigma(a_n)$ converges to $\lambda \in \mathbb{C}$, then $\lambda \in \sigma(a)$. Does the converse hold? So if $\lambda \in \...
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Proof: $C(X×Y)=C(X)⊗C(Y)$

Where I can find the proof of the following theorem: Let $X$ and $Y$ be compact Hausdorff spaces, $C(X)$ and $C(Y)$ the space of continuous functions on $X$ and $Y$ respectively, then we have $C(X×Y)=...
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Two question on a lemma about C*-algebra

I am reading Lin Hua xin's book "An introduction to the classification of amenable C*-algebras" and i am confused with the lemma 1.7.12 in this book. Lemma 1.7.12 Let $A$ be a C*-algebra and $f\in A^{...
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Seek results in group theory obtained by applping algebraic topology tools

After studying the first section, especially the section 1.3 (covering spaces) in Hatcher's book, it is obvious that many classical topics in group theory, especially in infinite group theory, have ...
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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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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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$\mathcal{H}_B$ is isomorphic to $B$ as Hilbert $B$-modules if and only if $B$ is a stable C*-algebra

Let $B$ be a C*-algebra. I want to prove that $\mathcal{H}_B$ is isomorphic to $B$ as Hilbert $B$-modules if and only if $B$ is stable, that is, $\mathcal{K} \otimes B$ is isomorphic to $B$ as C*-...
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Equivalence of categories ($c^*$ algebras <-> topological spaces)

I try to use a littlebit category theory to have a better overview of the results in the theory of $c^*$-algebras, but I really have to read an introduction to category theory because I know almost ...
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Examples of $C^*$-algebras in Noncommutative Geometry from A. Connes

Question I am working on $C^*$-algebras and I've been given Alain Connes's book Noncommutative Geometry. I am having troubles with understanding the examples on pages 91-93 (86-88 in the printed ...
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amenable groups versus amenable graphs

In operator algebras, one is often concerned with amenable groups, defined by one of many equivalent conditions. http://en.wikipedia.org/wiki/Amenable_group#Equivalent_conditions_for_amenability In ...
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Motivations for and connections between the topologies of Vietoris, Fell and Chabauty

My main interest is in the Chabauty topology on the space of closed subgroups of a locally compact topological group, merely out of curiosity. Wikipedia states "it is an adaptation of the Fell ...
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Comparison of positive elements and Hilbert C*-modules

I can't find a proof of facts like the following, which apparently are quite standard in the theory of C*-algebras. Let $\mathfrak A$ be any C*-algebra, and $a,b$ two positive elements in $\mathfrak ...
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Biduals generated by projections

This question is motivated by a similar question recently posed at MO: http://mathoverflow.net/questions/122091/masas-in-second-duals-of-banach-algebras In this setting, let $B$ be a Banach algebra ...
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194 views

Must-read papers in Operator Theory

I have basically finished my grad school applications and have some time at hand. I want to start reading some classic papers in Operator Theory so as to breathe more culture here. I have read some ...
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Commutant of bounded linear operators on a Hilbert space

Given a Hilbert space $H$, denote by $\mathcal{A}=\mathcal{B}(H)$ the C*-algebra of bounded linear operators on $H$. Denote further by $$\mathcal{B}(H)' := \{A\in \mathcal{B}(H) : [A,B]=0 \;\forall ...
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A problem on $C^\ast$-algebras and $W^\ast$-algebras

Let $I$ is a compact topological space, $m$ is a positive regular Borel measure. Then $L^\infty(m)$ is a standard example of commutative $W^\ast$-algebra (von Neumann algebra), but it is also a ...
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217 views

Confusion in Gelfand theorem in C*-algebra.

I am reading HX Lin's book, named "An introduction to the classification of amenable C*-algebras", I can not understand a corollary of Gelfand theorem(Corollary 1.3.6): If a is a normal element in a ...
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275 views

The free group $II_{\infty}$ factor isomorphism problem

Let $\Gamma$ be an infinite discrete group, and $H = l^{2}(\Gamma)$ the separable infinite dimensional Hilbert space. Let $\rho$ be the left regular representation of $\Gamma$ on $H$. Definition : ...
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Minimal projections in a C* algebra

Let $e$ be a projection in a C* algebra $A$. Is $eAe= \mathbb{C}e$ equivalent to the nonexistence of any projection in between $e$ and $0$? I know it is true if $A$ is a Von Neumann algebra because ...
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2answers
180 views

Why the weak * topology on the dual of a Banach space has the stronger meaning of locally compact

Let us say that for a Hausdorff topological space to be locally compact means that every point has a compact neighborhood. Why do locally compact have the property that if $x \in U$ and $U$ is open ...
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Why is $\overline{B(l^2)\odot B(l^2)}^{\| \enspace \|_{op}}\neq B(l^2\otimes l^2)?$

Let $B(l^2)$ be the $C^*$algebra of bounded linear operators on the sequence space $l^2$ and denote with $B(l^2)\odot B(l^2)$ the tensor product of $B(l^2)$ with itself, considered as a $*$algebra (...
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$C(S^1)$ 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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Show that a space X is homeomorphic to the space of multiplicative linear functionals

Let $\mathcal{A}=C(X,\mathbb{R})$ where $X$ is a compact Hausdorff space. Let $\hat{\mathcal{A}}$ be equal to the set of multiplicative linear functionals from $\mathcal{A}$ to $\mathbb{R}$. ...
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Irreducible representations and commutative C*Algebras.

If $A$ is a commutative C*-Algebra then also its representation $\pi(A)$ is commutative, and it's an operator C*-algebra. A representation is said to be irreducible if $\pi(A)$ does not commute with ...
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Why does exponentiating the derivative yield the shift operator?

If we formally exponentiate the derivative operator $\frac{d}{dx}$ on $\mathbb{R}$, we get $$e^\frac{d}{dx} = I+\frac{d}{dx}+\frac{1}{2!}\frac{d^2}{dx^2}+\frac{1}{3!}\frac{d^3}{dx^3}+ \cdots$$ ...
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A simple question about completely positive linear maps

Let $A$ be the C*-algebra and $M_{n}(A)$ be the C*-algebra of $n\times n$ matrices with entries in $A$. We use $(a_{ij})$ to denote the element of $M_{n}(A)$. My question is: For every $a\in A$, ...
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184 views

A simple question about *-homomorphism in C*-algebra

Let $A$ and $B$ be C*-algebra, $h\colon A\rightarrow B$ is *-homomorphism. If $a\in A_{\operatorname{sa}}$, then $\operatorname{sp}(h(a))\backslash \{0\}\subset \operatorname{sp}(a)\backslash\{0\}$. ...
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Questions about $B(H)$ and $B(H)/K(H)$ as Banach space

I am trying to investigate the relation between Uniformly Convexity and existence of Schauder Basis for a Banach space. I read in a Handbook article that $B(H)$ (the algebra of all bounded operators ...
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262 views

On the use of nets when defining operator topologies

Let's consider the strong operator topology and the weak operator topology on bounded operators of a infinite-dimensional Hilbert space $H$. When they define these operator topologies, some authors ...
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Polar decomposition of invertible elements in a unital C$ ^{*} $-algebra.

If $ A $ is a unital C$ ^{*} $-algebra and $ a $ is invertible, then $ a = u|a| $ for a unique unitary element $ u $ of $ A $. If $ \| a \| = \| a^{-1} \| = 1 $, what can you say about $ |a| $? I ...
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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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Gelfand-Naimark for $C^*$-categories

What is a reference for the following Theorem? If $A$ is a small $C^*$-category, then there is a faithful $C^*$-functor $A \to \mathsf{Hilb}$. $C^*$-categories with exactly one object are just $...
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Let $A$ be a $C^*$-algebra, $a \in A$ self adjoint

Question:Let $A$ be a $C^*$-algebra, $a \in A$ self adjoint. Suppose that the spectrum $\sigma(a)$ is an infinite set. Show that $A$ is infinite-dimensional. How can i prove it? I guess: Let $A$ be ...
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2answers
482 views

Analytic Vectors (Nelson's Theorem)

Is there a (simple) proof for Nelson's theorem that a symmetric operator is essentially selfadjoint if it contains a dense subset of analytic vectors?
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The canonical surjection between the full and the reduced group C^*-algebras

This might be an incredible easy question -- since any reference I've found state it as obvious -- but anyway: Given a group $G$, I can construct the full group-$C^*$-algebra $C^*(G)$ be completing $...
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an invariant of $C^{*}$ algebras

consider the following property (invariant) for complex $C^{*}$ algebras: "$T(x)=x^{*}$ is the only non zero $\mathbb{R}$-linear map on $A$ which satisfies $T(x)T(y)=T(yx)$." Questions: 1)Some ...
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A theorem about conditional expectation in C*-algebra

Definition 1. Let $B\subset A$ be C*-algebra. A projection from A onto B is a linear map $E: A \rightarrow B$ such that $E(b)=b$ for every $b\in B$. A conditional expectation from A onto B is a ...
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equality of two operators…

Please help me with the following problem( give some hints or references): Let $X$ be a Banach space and $B(X)$ be the algebra of bounded linear operators on $X$. Suppose that $A$ and $B$ are two ...
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347 views

Learning roadmap for Non-commutative Geometry [closed]

I am interested in learning Non-commutative geometry and K-theory of operator algebras. Please suggest a learning roadmap for this subject. My present knowledge of Measure theory & Functional ...
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245 views

A question on multiplicative linear functional on Banach algebra.

I am reading a book about C*-algebra. But i am confused with some of its content. It says Assume $A$ is a non-unital C*-algebra and $\tilde{A}$ is its unitization (the elements of the form $a+\lambda$...
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107 views

Algebra (Not *)-Isomorphisms of von Neumann algebras

Let $A$ and $B$ be any two infinite-dimensional von Neumann algebras, they are operator algebras with operator composition as the multiplication and as infinite dimensional vector spaces they're ...
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The span of the orthorgonal projections is norm dense in $B(H)$

This is a question in my functional analysis book. How to use the spectral theorem to prove that the span of the orthogonal projections is norm dense in $B(H)$?
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Abelian von Neumann Algebras on non-separable Hilbert spaces

Is there a classification of Abelian von Neumann algebras on non-separable Hilbert spaces? For a classification of Abelian von Neumann algebras on separable Hilbert spaces, see this link.
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Strong convergence of projections in $B(H)$

Let $\{e_{kj}\}$ be the canonical matrix units in $B(H)$, with $H$ separable. Define projections $q_k$ by $$ q_k=\sum_{n=1}^ke_{nn}. $$ Let $\{p_1,p_2,\ldots\}\subset B(H)$ be a sequence of ...