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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Ordering: Definition

This was a real question! Given a unital C*-algebra $1\in\mathcal{A}$. For $A\in\mathcal{A}$ denote its spectrum by $\sigma(A)$. Consider the selfadjoints: ...
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31 views

An strange trace class operator

Assume $H$ is a non separable Hilbert space. 1- Let $\{a_n\}$ be a sequence of bounded linear operators on $H$. Any operator $a_n$ induces the bounded linear functional $x\to tr(xa_n)$ on $L^1(H)$, ...
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1answer
44 views

Showing that there exists a functional on $l^\infty$ whose value lies between the infimum and supremum of a sequence-is my solution correct?

I'm trying to show that there exists a functional in $l^\infty$ whose value lies between the infimum and supremum of a sequence. That is, there exists a functional $\phi:l^\infty\longrightarrow ...
3
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1answer
53 views

C*-algebraic intrinsic definition for compactness of an operator?

Some properties of operators (normal, self adjoint, hermitian) have intrinsic definitions for any element of a $C^*$-algebra. Is there such definition for compact operators? Equivalently: Let ...
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1answer
117 views

Why the disk algebra is not a C* algebra.

I'm trying to figure out why the set of bounded analytic functions on the unit disk, A(D), is not a C* algebra. The norm is the sup norm and the involution is $f(z) \to \overline{f(\bar z)}$. I want ...
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303 views

Does category theory help in operator algebras?

I'm currently studying the basics of Banach and $C^*$-algebras. Almost all the proofs i've seen so far are very simple but some of them are extremely tricky (in my opinion). This tricky interplay ...
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59 views

Operator realizations of algebras

I want to realize the algebra $A_q(\tilde{S}^{n-1})$ as introduced in the acticle of Dijkhuizen and Noumi (http://arxiv.org/pdf/q-alg/9605017v1.pdf) as bounded operators on a Hilbert space $H$. Can ...
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Norm of a Positive Operator

For a positive operator $A\in B(\mathcal{H})$ on a complex Hilbert space $\mathcal{H}$, I want to prove that $\|A\|=\sup\{\lambda: \lambda \in \sigma(A)\}$, where $\sigma(A)$ is the spectrum of $A$. ...
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44 views

$\mathcal{A}+K$ is norm-closed where $\mathcal{A}$ is a $C^*$-algebra and $K$ is the compact operators.

Let $\mathcal{A}\subset B(H)$ be a unital $C^*$-algebra and let $K$ be the closed ideal of compact operators. I need to show that $\mathcal{A}+K$ is also a $C^*$-subalgebra of $B(H)$. I am stuck at ...
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25 views

Maximal ideal space and quotient space in abelian Banach algebra

I have a short question regarding operator algebras. Given an abelian Banach algebra $\mathcal{A}$. Assume that $\phi \in \big\{ \phi : \phi \text{ is a non-zero linear multiplicative functional} ...
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1answer
52 views

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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53 views

Unit of a purely infinite, simple C*-algebra

Suppose that we have a purely infinite, simple C*-algebra with unit $1$. Can we find two projections $p,q$ both equivalent to the identity such that $1=p+q$ and $pq=0$? Well, there are two ...
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2answers
33 views

A question about corners of $C^\ast$-algebras

Let $\mathcal{A}$ be a $C^\ast$-algebra, $p\in M_n(\mathcal{A})$ a projection, is there a $k\in\mathbb{N}$ such that $pM_n(\mathcal{A})p\cong M_k(\mathcal{A})$ ? Thanks a lot!
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1answer
38 views

States on a non-unital $C^*$-algebra

Let $\mathcal{A}$ be a unital $C^*$-subalgebra of $B(H)$. Then the definition $\phi(a):=\langle ah,h \rangle$ for a fixed $h\in H, \|h\|=1$ and for all $a\in\mathcal{A}$ defines a state on ...
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1answer
79 views

When can $*$-algebras be turned into $C^*$-algebas?

Let $A$ be a (not necessarily unital) complex $*$-algebra, i.e. an algebra over $\mathbb{C}$ together with an involution $*: A \to A$. There exists at most one norm on $A$ turning $A$ into a ...
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1answer
36 views

Matrices over the Cuntz algebra

Consider the Cuntz algebra $O_2$. Is it true that $M_2(O_2)$ is isomorphic to $O_2$? I was trying to show that is impossible but now I am not sure.
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star-modules and star-algebras

Is the following concept already known, perhaps with a different name? Let $R$ be a commutative ring and $* : R \to R$ be an involutive homomorphism (the most typical case is $R=\mathbb{C}$ and $* = ...
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29 views

Subordinate projections: Transitivity

Let $A$ be a $C^*$-algebra. If necessary, let us assume that $A$ is a von Neumann algebra. For projections $p,q \in A$ one writes $p \prec q$ if $p$ is Murray-von Neumann-equivalent to a subprojection ...
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48 views

a point on trace class operators

Assume $H$ is separable Hilbert space and fix an orthonormal basis $\{e_n\}_1^{\infty}$. Let us denote $p_n$ by the projection onto the subspace generated by $\{e_1\cdots,e_n\}$. Let $a$ be a ...
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85 views

Arzelà–Ascoli theorem for operators

If you have a net of continuous linear operators between reasonable spaces (complete, at least), does there exist Arzelà–Ascoli-like theorems giving convergent subnets? I believe that it should be ...
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1answer
104 views

Characterization of normal operators on Hilbert space as function of a self-adjoint operator

My question : Suppose T is a normal operator on a Hilbert space H. Show that there exists a self-adjoint operator S on H such that T=f(S), where f is continuous function from spectrum of S into S. My ...
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1answer
39 views

Normal element is sum of four commuting positive elements

I am stuck with the following problem : Every normal element in a $C^*$ algebra can be written as a linear combination of four commuting positive elements. I had tried along the following lines. ...
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Introductory books for ‎ ‎$\frak{E}_p(I)$

Are there any good books different from abstract harmonic analysis by hewitt to study ‎$\frak{E}_p(I)$. where ‎$\frak{E}_p(I)$ is: ‎Let $I$ be an arbitrary index set‎. ‎For each $i\in I$ let $H_i$ ...
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1answer
30 views

How to handle direct sums and unitizations of $L^p$ operator algebras?

Let $p\in[1,\infty)$. An $L^p$ operator algebra refers to a Banach algebra that is isometrically isomorphic to a closed subalgebra of $B(L^p(X,\mu))$ for some ($\sigma$-finite) measure space ...
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Can $AB-BA=I$ hold if both $A$ and $B$ are operators on an infinitely-dimensional vector space over $\mathbb C$?

Of course, it can't hold if operators are over finite-dimensional spaces, as is evident from trace considerations. Can it be true for infinite-dimensional spaces? I think not, but I don't see how we ...
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1answer
33 views

Can someone help me to give some hints? Left Hilbert-$C_0(T,K(H))$ module $C_0(T,H)$

I tried to prove example 3.4 from the book Morita Equivalence and Continuous-Trace C$^*$-Algebras by Iain Raeburn and Dana P. Williams, but I get uneasy with notations and ideas. Let me restate my ...
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91 views

What is an algebra and what is it's representation?

Heyho, i've kind of got an understanding problem what exactely an algebra and especially it's representation is. In my case it is said, that the relation $R_{12}(u-v) (L(u) \otimes \mathrm{I}) \; ...
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2answers
127 views

What, and how can, topological invariants can be computed from a space's algebra of functions?

The Gelfrand duality says that the category of locally compact Hausdorff spaces (with proper continuous functions) is equivalent to the category of commutative $C^*$ algebras (with proper ...
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1answer
95 views

Power series expansion of an Operator.

I've been reading a paper called "Separation of variables for the quantum $Sl(2,R)$ spin chain" in which the author at one point does a power series expansion I do not understand. The problem is this ...
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Clarification on Wolfram Mathworld's explanation of the connection between Gelfand Transform and Fourier Transform

http://mathworld.wolfram.com/GelfandTransform.html In the definition, what does $x$, $\hat x(\phi)$, and $\phi$ represent exactly if we were to consider definition of the Fourier transform? Can ...
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1answer
101 views

Is there a $C^{*}$ algebra with these properties

Is there a unital C* algebra A which is NOT simple but satisfies the following two conditions: 1)A has trivial center 2)A has a faithful trace such that every zero trace element lies in the closure ...
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1answer
30 views

Ordering: Normalized

Given a Hilbert space $1\in\mathcal{A}$. Denote selfadjoints: $$\mathcal{S}:=\{A\in\mathcal{A}:A=A^*\}$$ Introduce an order: $$A\leq A':\ \ \sigma(A'-A)\geq0$$ Regard a projection: $$P\neq0:\quad ...
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Weak (operator) null sequence is bounded and pointwise convergent to zero

I was reading Diestel book (Absolutely Summing Operators) and it says: "(...) let $(f_n)$ be any weak null sequence in $\mathcal{C}(K)$. Then $(f_n)$ is bounded and converges pointwise to zero." I ...
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1answer
50 views

Operators whose spectrum has a finite number of connected component

Assume that $H$ is a separable Hilbert space. Let $Q$ be the set of all operators$T \in B(H)$ such that the spectrum of $T$ has a finite number of connected component. Is $Q$ a subvector space ...
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32 views

Derivation into dense ideal of Banach algebras

Let $A$ be a Banach algebra and $I$ be an ideal of $A$. A derivation $D:A\to I$ is a linear bounded map, with the following property: $$D(ab)=aD(b)+D(a)b,\qquad a,b\in A.$$ Suppose that $I$ is dense ...
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1answer
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A question about the weak operator topology of $B(H)$

Let $H$ be a Hilbert space and $V$ a dense subspace. Let $A$ be a $*$-subalgebra of $B(H)$. The weak closure of $A$ is, by definition, the space of all $u\in B(H)$ satisfying: For every ...
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1answer
100 views

Normal Operators: “Zorn”

Problem Given a Hilbert space $\mathcal{H}$. Normal Operators: $$\mathcal{N}(\mathcal{H}):=\{N:N^*N=NN^*\}$$ (Possibly unbounded!) Borel Calculus: ...
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1answer
33 views

Quasi ideal sequence in $B(H)$

According to comments by Hamza I revise the question. Let $H$ be an infinite dimensional separable Hilbert space. Is there an increasing sequence of subvector spaces $V_{1} \subsetneq V_{2} ...
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1answer
56 views

Distributivity of projective tensor product over direct sum

Let $I$ is a non-empty set and $\{A_i\}_{i\in I}$ is a family of Banach algebras and $B$ is a Banach algebra. Define $$\ell^1-\oplus_{i\in I}A_i=\{a=\{a_i\}_{i\in I}: \|a\|_1=\sum_{i\in ...
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1answer
36 views

composition and strong limits of completely positive maps is completely positive

I have two claims about completely positive maps. Let $X$, $Y$, $Z$ be $C^\ast$-algebras. 1) Let $f:X\to Y$ and $g:Y\to Z$ be completely positive maps. I want to know, why $g\circ f$ is completely ...
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1answer
50 views

the $C^\ast$-algebra $M_n(A)$, understanding the $C^\ast$-norm on $M_n(A)$

Let $A$ be a $C^\ast$-algebra. I want to understand $M_n(A)$, the vector space of $n\times n$-matrices with entries in $A$, as a $C^\ast$-algebra. On $M_n(A)$ you can define an involution ...
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1answer
59 views

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 ...
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Gaussian unitary dilation of Gaussian channels

I am starting with a few definitions. All these are standard and can be accessed from some quantum information or quantum physics books, for instance the books by Holevo or Parthasarathy. The question ...
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Semiprimitivity of second dual of semiprime Banach algebras

Let $A$ be a Banach algebra. Then $A^*$ is right Banach $A$-module with product $\langle b,f.a\rangle=\langle ab,f\rangle$ for every $a,b\in A, f\in A^*$. Define $\langle a,F*f\rangle=\langle ...
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1answer
32 views

Weakly compact operator with different domains

Let $A$ be a Banach algebra. Suppose that $e\in A$ such that $e^2=e$ and $eAe$ is division algebra(i.e., $eAe$ is unital and every element of $eAe$ has inverse in $eAe$). Define $T_e:A\to A$ with ...
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2answers
53 views

Projections: Orthogonality

Given a unital C*-algebra $1\in\mathcal{A}$. Consider projections: $$P^2=P=P^*\quad P'^2=P'=P'^*$$ Order them by: $$P\perp P':\iff\sigma(\Sigma P)\leq1\quad(\Sigma P:=P+P')$$ Then equivalently: ...
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Meaning of kernel

If something is the 'kernel' of a transformation, say $K(x,x')$, does it mean I should take the integral $$\int K(x,x') f(x')dx'$$ There are many different meanings of kernel and I did not see their ...
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Uniqueness of c.p.c. order zero extensions

I have a question about a passage in the proof (on page 316) of proposition 3.2 in this paper http://wwwmath.uni-muenster.de/42/fileadmin/Einrichtungen/mjm/vol_2/mjm_vol_2_14.pdf. My question is ...
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55 views

Compactum of Banach algebra

I need an example of Banach algebra $A$ and a left non-trivial closed ideal $I$ with all of following properties: There exists a bounded approximate identity in $I$ for $I$ i.e., a net ...
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1answer
42 views

Relation between tracial states on von Neumann algebras and their GNS representations

Let $M$ be a von Neumann algebra acting on a Hilbert space $H$, and let $\tau$ be a faithful tracial state on $M$. What is the relation between the GNS representation of $(M,\tau)$ and the original ...