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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881 views

Are commutative C*-algebras really dual to locally compact Hausdorff spaces?

Several online sources (e.g. Wikipedia, the nLab) assert that the Gelfand representation defines a contravariant equivalence from the category of (non-unital) commutative $C^{\ast}$-algebras to the ...
19
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1answer
408 views

Why study operator spaces?

I'm currently enrolled in an operator spaces course and I'm finding it difficult to understand why we study them in the first place. Functional analysis is motivated well enough for me and even though ...
18
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8answers
1k views

Reference for spectral sequences

What are good expositions of spectral sequences, which include a thorough introduction to the topic as well as the most important examples of applications - maybe with an emphasis an topological ...
15
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1answer
565 views

Motivation for abstract harmonic analysis

I am reading Folland's A Course in Abstract Harmonic Analysis and find this book extremely exciting. However it seems Folland does not give many examples to illustrate the motivation behind much of ...
13
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1answer
344 views

How does $\sigma(T)$ change with respect to $T$?

Consider $\sigma$ as a mapping which maps $T\in\mathcal{L}(X)$ to $\sigma(T)$, the spectrum of $T$, a compact set in the complex plane. I wonder whether there is some result concerning how ...
12
votes
2answers
789 views

Is there an algebraic homomorphism between two Banach algebras which is not continuous?

According to wikipedia, you need the Axiom of Choice to find a discontinuous map between two Banach spaces. Does this procedure also apply for Banach algebras yielding a discontinuous multiplicative ...
11
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2answers
694 views

$C^*$-algebra which is also a Hilbert space?

Does there exist a nontrivial (i.e. other than $\mathbb{C}$) example of a $C^*$-algebra which is also a Hilbert space (in the same norm, of course)? For $\mathbb{C}^n$ with $n > 1$ the answer is ...
11
votes
1answer
432 views

Renorming $\mathcal{B}(\mathcal{H})$?

Consider the Banach space of all bounded operators $\mathcal{B}(\mathcal{H})$ on a (separable if you wish) Hilbert space $\mathcal{H}$ with the operator norm. Can we renorm this space to a strictly ...
10
votes
3answers
674 views

Applications of Banach Algebras and Operator Algebras

I am trying to learn operator algebra theory (I am tempted to start with Douglas' "Banach Algebra Techniques in Operator Theory"). One aspect that I am curious about is whether there are significant ...
10
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1answer
218 views

Maximal ideal space of $c_{\mathcal{U}}$

Let $\mathcal{U}$ be an filter over $\mathbb{N}$. Define $$c_{\mathcal{U}} = \{{(x_n)\in \ell_\infty\colon \lim_{\mathcal{U}, n}x_n =0\}},$$ which is a C*-algebra. Is there an accessible topological ...
9
votes
2answers
235 views

Why can we classify the W*algebra?

Many operator algebra books discuss the classifiation of W*algebra(von Neumann algebra),but not the C*algebra,why? I think a direct reason is that we have the projection comparison theorem in the ...
9
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1answer
597 views

What is the use of Spectral Theorem?

Obviously the version for compact and self-adjoint linear operators on Hilbert Spaces is very useful since it decomposes the operators into orthogonal projections. However, the following more general ...
9
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1answer
347 views

Maximal ideals and maximal subspaces of normed algebras

This is a kind of "prove or give a counter-example" question, and I'm having some difficults with it: By a maximal ideal $I$ of an algebra $A$, we mean an ideal $I\neq A$ which is not properly ...
9
votes
1answer
182 views

Do we have Maximal Abelian Algebras (MAAs)?

Let $\mathcal{H}$ be a Hilbert space and $B(\mathcal{H})$ the algebra of bounded linear operators on $\mathcal{H}$. A MASA $\mathcal{M}$ is a subalgebra of $B(\mathcal{H})$ that is abelian and ...
8
votes
1answer
389 views

reference for operator algebra

I am taking a course on operator algebra this semester. My instructor has suggested a reference "Kadinson and Ringrose." Are there any other good/standard references for this subject that I can look ...
8
votes
2answers
196 views

Traces on separable simple $C^{\ast}$- algebras

What is an example of a separable, simple $C^{\ast}$-algebra that admits two different tracial states? EDIT: Julien has pointed to a number of avenues to answer this question. If anyone has an ...
8
votes
1answer
422 views

Ideals in $C(X)$

Let $X$ be a Hausdorf Compact topological space. Please help me to show, for the purpose of understanding an example in some of my lecture notes, that the closed ideals in $C(X)$ are of the following ...
7
votes
2answers
128 views

Recovering a group from its C*-algebras and group algebra

Let $G$ and $H$ be locally compact groups. Does anyone know the answers to these questions? Is it true that: if $C^*(G)$ and $C^*(H)$ are $*$-isomorphic, then $G\cong H$? if $C_r^*(G)$ and ...
7
votes
1answer
135 views

Ideals in $B(H)$ are self-adjoint

It is known that every (closed two-sided) ideal in a $C^{*}$-algebra is self-adjoint. The proofs that I've seen involve functional calculus and approximate units. I am wondering whether there is a ...
7
votes
2answers
466 views

A question about pure state

For every unit vector $x$ in a Hilbert space $H$,let $F_x$ be the linear functional on $\mathcal B(H)$ (bounded linear operators) defined by $F_x(T)=(Tx,x)$. Prove that each $F_x$ is pure state and ...
7
votes
2answers
227 views

There are 3 points in the spectrum of some self-adjoint element of a non-unital C*-algebra.

Let $A$ be a non-unital C*-algebra. I would like to know a simple way to show that $A$ contains a self-adjoint element whose spectrum has at least $3$ elements. Note that the spectrum of an ...
7
votes
2answers
231 views

Gelfand Naimark Theorem

The commutative Gelfand-Naimark theorem tells us that every unital commutative C* algebra is isometrically isomorphic to the space of continuous functions on its maximal ideal space. The non- ...
7
votes
1answer
693 views

Property of partial traces

Consider the Kronecker product of $A \in M_m, B \in M_n$: $A \otimes B = \left( \begin{matrix} a_{11}B&...&a_{1m}B\\ \vdots&\ddots\\a_{m1}B&...&a_{mm}B \end{matrix} \right)$ $A ...
7
votes
1answer
49 views

simple connectedness and abelian C* algebras

From Gelfand-Neumark Theorem, we know that topological properties of a compact Hausdorff space $X$ are encoded in the abelian $C^*$-algebra of continuous complex-valued functions on $X$ (with $||f||= ...
7
votes
1answer
161 views

When is a $*$-homomorphism between multiplier algebras strictly continuous?

The strict topology on the multiplier algebra $M(A)$ of a C*-algebra $A$ is that generated by the seminorms $$ x\mapsto \| ax \|\qquad x\mapsto\| xa \| \qquad (x\in M(A), a\in A) $$ Whereas a ...
7
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0answers
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+500

When two projections in a C*-algebra are “almost” Murray-von Neumann equivalent, they are equivalent

Let $A$ be a C*-algebra and $p,q \in A$ be projections. Assume there is an element $a\in A$ such that $\|aa^*-p\|<\frac{1}{4}$ and $\|a^*a-q\|<\frac{1}{4}$. Then there is a partial isometry $v$ ...
7
votes
0answers
127 views

K-theory for non-separable C*-algebras

Let $\kappa$ be an uncountable cardinal. What is the K-theory for the C*-algebras $\mathcal{K}(\ell_2(\kappa))$ and $\mathcal{B}(\ell_2(\kappa))$, of, respectively, compact and bounded operators on ...
6
votes
2answers
398 views

Non-$C^{*}$ Banach algebras?

It suddenly occurred to me almost every Banach algebra I know is actually a $C^{*}$ algebra. Several kinds of function algebras are definitely $C^{*}$ algebras. So is the matrix algebra. Although one ...
6
votes
2answers
232 views

Why are compact operators 'small'?

I have been hearing different people saying this in different contexts for quite some time but I still don't quite get it. I know that compact operators map bounded sets to totally bounded ones, that ...
6
votes
1answer
183 views

equivalent? algebraic definition of a partial isometry in a C*-algebra

An element $a\in\mathfrak{A}$ (unital C*-algebra) is a partial isometry if $a^*\cdot a $ is projection. Can one recover the equivalent caracterizations of a partial isometry in ...
6
votes
1answer
306 views

Cube root in $ C^{*}$-algebra.

Let $A$ be a $C^*\text{-algbera}$ and $x\in A$. I'm trying to show thata)for $0<\alpha<\frac{1}{2}$, there exists $u\in A$ with $x=u(x^*x)^{\alpha}$ and $u^*u=(x^*x)^{1-2\alpha}$. b) there ...
6
votes
1answer
348 views

Matrices with entries in $C^*$-algebra

Let $\mathcal{A}$ be a $C^*$-algebra. Consider vector space of matrices of size $n\times n$ whose entries in $\mathcal{A}$. Denote this vector space $M_{n,n}(\mathcal{A})$. We can define involution on ...
6
votes
1answer
159 views

In how far are CCR and GCR C*-algebras interesting?

I have some basic familiarity with C*-algebras (from the mathematical physics side - Bratteli-Robinson), and while having a look at Arveson's "Invitation to C* algebras", I came across so-called CCR ...
6
votes
1answer
646 views

Some examples in C* algebras and Banach * algebras

I would like an example of the following things. A Banach * algebra that is not a C* algebra for which there exists a positive linear functional (it takes $x^*x$ to numbers $ \geq 0$) that is not ...
6
votes
1answer
271 views

Visualize operator algebras?

It seems to me that to study mathematics is to convert the abstract language into diagrams, graphs and images. It does depend on the subject how much this technique can ease the struggle yet most of ...
6
votes
1answer
239 views

Weak* operator topology and finite rank operators

We will say that ${T_i}\subset B(X,Y^*)$ converges to $T$ in W*-operator topology if $T_i(x)\rightarrow T(x)$ in W*-topology of $Y^*$( $\forall y\in Y \langle T_i(x),y\rangle \rightarrow \langle ...
6
votes
1answer
252 views

Homomorphic conditional expectations?

To clarify, I mean "conditional expectation" in the sense of $C^*$-algebras (a completely positive projection of norm 1, equivalently, a completely positive linear map onto a $C^*$-subalgebra which is ...
6
votes
2answers
119 views

Are self-inverse operators normal?

Let $\mathcal{H}$ be an Hilbert space. Consider a bounded Operator $T:\mathcal{H}\to \mathcal{H}$. Suppose $TT=1$, does it hold, that $T^{*}T=TT^{*}$? If so, how does one show this? If not, what kind ...
6
votes
1answer
300 views

Weak-* continuity of the adjoint map on a $W^*$-algebra

Let $\mathcal{M}$ be a $W^*$-algebra, i.e. a $C^*$-algebra with a Banach space predual $\mathcal{M}_*$. I'm trying to show that the adjoint map $x \mapsto x^*$ on $\mathcal{M}$ is weak-* (aka ...
6
votes
1answer
147 views

Question on irreducible representation of a Banach algebra

Let $\mathcal A$ be a Banach algebra over $\mathbb{C}$, $\mathcal X$ a irreducible left $\mathcal A$-module. If $x,y \in \mathcal X$ are linearly independent, there exists an element $a\in\mathcal A$ ...
6
votes
1answer
200 views

$\mathcal{K}(L^2(\mathbb{R}^m \times \mathbb{R}^n)) = \mathcal{K}(L^2(\mathbb{R}^m)) \otimes \mathcal{K}(L^2(\mathbb{R}^n))$?

QUESTION: Is it true that for the algebra of compact operators: $\mathcal{K}(L^2(\mathbb{R}^m \times \mathbb{R}^n))$ is as a $C^{\ast}$-algebra isomorphic to $\mathcal{K}(L^2(\mathbb{R}^m)) \otimes ...
6
votes
1answer
343 views

Abstract Von Neumann Algebras

I have just read this question Is a von Neumann algebra just a C*-algebra which is generated by its projections? and am wondering about Robert Israel's answer when he says that a subalgebra of $C(X)$ ...
6
votes
0answers
254 views

Combinatorial total space for finitely generated torsion-free groups?

Motivation: I'm an operator algebraist and I'm looking for an answer to the main question in order to build non-trivial spectral triples for a class (as large as possible) of discrete groups. $\to$ ...
6
votes
0answers
151 views

How to prove this element is strictly positive? [duplicate]

Let $A$ be a $C^*\text{-algebra}$ and $A_+$ denote the positive elements. An element $a\in A_+$ is called strictly positive if $\overline{aAa}=A$. Want to prove: if $(e_n)$ is an approximate identity ...
6
votes
0answers
85 views

Products in $C^*$-algebra $K$-theory

Let $A_1$ and $A_2$ be unital $C^*$-algebras. If $p_1 \in M_{n_1}(A_1)$ and $p_2 \in M_{n_2}(A_2)$ are projections then $p_1 \otimes p_2 \in M_{n_1 n_2}(A_1 \otimes A_2)$ is also a projection, ...
5
votes
2answers
896 views

Is a von Neumann algebra just a C*-algebra which is generated by its projections?

von Neumann algebras have the nice property that they are generated by their projections (the elements satisfying $e = e^{\ast} = e^2$) in the sense that they are the norm closure of the subspace ...
5
votes
1answer
674 views

Gelfand Topology and C*-algebras

Before we start here some notations to have no confusion: Suppose $A$ is a commutative $C^*$-algebra with unit. $\Sigma(A)$ is the Gelfand spectrum, given by all linear maps ...
5
votes
1answer
208 views

Is the centre of a C*-algebra a sub-C*-algebra?

I believe that the answer is affirmative and I would be grateful to any comments on my attempt (see below) of proving this. Let $A$ be a C*-algebra and denote by $Z(A)$ the centre of $A$. First of ...
5
votes
2answers
162 views

Duals via a Bilinear map

Let $E$ and $F$ be normed vector spaces. Then if $B$ is a bounded bilinear form on $E \times F$ then every $y \in F$ defines a bounded linear functional $f_y$ where $f_y(x)=B(x, y) \forall x \in E$. ...
5
votes
1answer
804 views

Trace of an operator

Suppose $\rho$ is a positve trace-class operator and $x$ is positve bounded operator on a Hilbert space $\mathcal{H}$, I am unable to prove that trace of $\rho x$ is positive, where trace($x$):= $\sum ...