1
vote
0answers
30 views

Where is the most clear and concise exposition of the spectral theorem for self-adjoint operators on Hilbert space?

This question is certainly subjective, which may warrant votes to close. I'm simply looking to find the "best" written exposition of the spectral theorem for possibly unbounded self-adjoint operators ...
2
votes
1answer
18 views

Commutative subspace lattice

I have seen an article in which there is an algebra which was named CSL-algebra (Commutative Subspace Lattice). This algebra is about projection on Banach algebra? I couldn't find any good source to ...
1
vote
0answers
83 views

Groupoid $C^*$ algebra of product groupoid

Let $G$ and $H$ be locally compact (Hausdorff, second countable) groupoids with Haar systems $\mu$ and $\nu$, respectively. Is it true then that the (full) groupoid $C^*$-algebras satisfy $$ ...
2
votes
1answer
24 views

Is the von neumann algebra of locally compact amenable group hyperfinite?

Let $G$ be a discrete group and $\mathcal{L}(G)$ the associated von Neumann algebra. It is well known that $G$ is amenable if and only if $\mathcal{L}(G)$ is hyperfinite. Does there exist a ...
0
votes
1answer
32 views

Cross Product Algebras references

Can someone give some references to introductory books or online notes about group algebras and cross-product algebras ? I've already searched on Google (but only for some online notes). The purpose ...
4
votes
1answer
67 views

MASAs of C* algebras

While studying the $C^*$-algebraic formulation of the recently solved Kadison-Singer problem, I was wondering about maximal abelian subalgebras: Let $\mathcal{A}$ be a unital C* algebra. There seems ...
2
votes
1answer
126 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 ...
1
vote
1answer
105 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 : ...
8
votes
1answer
234 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 ...
2
votes
1answer
85 views

Is there an irreducible, noncompact commuting, nonnormal operator, with spectrum strictly continuous?

Let $H$ be an infinite dimensional separable Hilbert space. Definition: The commutant $\mathcal{S}'$ of a subset $\mathcal{S} \subset B(H)$ is $ \{A \in B(H) : AB=BA \ , \ \forall B \in \mathcal{S} ...
8
votes
2answers
172 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 ...
10
votes
3answers
385 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 ...
5
votes
1answer
76 views

characters of a $C^*$-algebra

I have read that a state $\rho$ on a unital $C^*$-algebra $A$ is a character (i.e. multiplicative) if and only if, for all unitary $u\in A$, $|\rho(u)| = 1$. Is there an easy proof, or can someone ...
4
votes
1answer
168 views

Quotients of C*-algebras

It is known that every unital separable C*-algebra is a quotient of the full group C*-algebra $C^*(F_I)$, where $F_I$ is the free group generated by some index set $I$. Can we drop the ...
2
votes
1answer
78 views

A reference request for sums of $C^*$-algebras

Does anyone know where I can find a reference for the following well-known fact: Let $(X_i)_{i\in I}$ be a family of compact Hausdorff spaces and let $X$ be the disjoint sum of all $X_i$s. Then ...
1
vote
0answers
63 views

Containment of an element to an operator system

This question will probably appeal to people in operator systems theory as it is very much related. However, I'm interested in down-to-earth concrete systems with finite dimensional Hilbert space ...
1
vote
0answers
64 views

Positive maps on $\mathcal{B}(\mathcal{H})$ to itself

Let us consider the set of positive maps $\phi:\mathcal{B}(\mathcal{H})\rightarrow \mathcal{B}(\mathcal{H})$ ($\mathcal{H}$ is Hilbert space). Can we characterize all the maps which satisfies the ...
2
votes
1answer
186 views

Extension of Choi's theorem on extreme completely positive maps

In this paper Man-Duen Choi gave a criteria for a completely positive map to be extreme. For convenience I am writing it below. Let $\phi:\mathcal{M}_n\rightarrow\mathcal{M}_m$. Then $\phi$ is ...
3
votes
1answer
112 views

Necessary and sufficient conditions for $G$ to have $A(G)$ isomorphic to an operator algebra?

Let $G$ be a locally compact group. We denote by $A(G)$ the Fourier algebra of $G$. An operator algebra is a closed subalgebra of $B(H)$ where $H$ is a Hilbert space. What are the necessary and ...
2
votes
0answers
52 views

Quotients of the CAR algebra

Recently, I heard about the following theorem: each nuclear separable operator space is a completely bounded quotient of the CAR algebra. Yet, I have no idea who and where proved this theorem ...
13
votes
1answer
327 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 ...
4
votes
0answers
103 views

Direct limits of completely positive maps on $C^*$-algebras vs. operator systems

I believe I've heard, as part of the "lore," that the category (operator systems, completely positive maps) has direct limits, whereas the category ($C^*$-algebras, completely positive maps) does not. ...
1
vote
1answer
45 views

Where can I find an English version of this paper by Gel'fand and Raikov?

It is titled 'Irreducible unitary representations of locally bicompact groups' and the original version is in Russian. Google scholar shows it has been translated into English and once pubulished in ...
15
votes
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 ...