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 ...
13
votes
1answer
277 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 ...
13
votes
3answers
324 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 ...
12
votes
7answers
683 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 ...
11
votes
2answers
453 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
votes
1answer
322 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 ...
9
votes
0answers
133 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 ...
8
votes
2answers
374 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 ...
8
votes
1answer
313 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 ...
8
votes
1answer
141 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 ...
7
votes
1answer
237 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 ...
7
votes
1answer
231 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
144 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
1answer
100 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
votes
0answers
227 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 ...
6
votes
2answers
84 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 ...
6
votes
1answer
170 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 ...
6
votes
1answer
347 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
124 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 ...
6
votes
1answer
167 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
233 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
162 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 ...
5
votes
1answer
125 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
1answer
121 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 ...
5
votes
2answers
137 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 ...
5
votes
1answer
41 views
States on a C*algebra
A state $\omega$ on a unital $C^*$ algebra $A$ is called definite at $a\in A$ self-adjoint if $\omega(a^2)=\omega(a)^2$.
I proved that if we have such a definite state at $a$, then for all $b\in A$ ...
5
votes
1answer
52 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 ...
5
votes
0answers
53 views
how to prove this element is strictly positive?
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 ...
5
votes
0answers
79 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 ...
5
votes
1answer
150 views
Separating vectors for $C^*$-algebras
Let $A$ be a C$^*$-algebra, concretely acting on a Hilbert space $H$. Suppose that $\xi_0\in H$ is cyclic and separating for $A$ (that is, the map $A\rightarrow H, a\mapsto a(\xi_0)$ is injective ...
4
votes
4answers
192 views
Applications of Operator Algebras to modern physics
I think that recently I've started to lean in my interest more towards operator algebras and away from differential geometry, the latter having many applications to physics. But while taking physics ...
4
votes
2answers
122 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 $f_y(x)=B(x, y) \forall x \in E$. ...
4
votes
2answers
278 views
An application of Artin Wedderburn and Schur's Lemma
In an operator algebras class, the professor said the following.
Let $G$ be a finite group, and consider the complex group algebra it generates. Using the following 3 facts, we're supposed to see ...
4
votes
1answer
95 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 ...
4
votes
1answer
97 views
States and positive elements in $C^*$-algebras
Let $A$ be a unital $C^*$-algebra and $w$ be a state (i.e a positive linear functional such that $\|w\|=w(1_A)=1$. I'm trying to prove the following:a) if $a$ is selfadjoint and $w(a^2)=w(a)^2$ then ...
4
votes
2answers
105 views
How to prove compactness of matrix convex sets?
I am reading a paper - "the Krein Milman theorem in Operator Convexity"; and the third section there deals with compact matrix convex sets. The first example there states that the matrix interval ...
4
votes
1answer
160 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 ...
4
votes
1answer
33 views
When is the image of a GNS representation WOT-dense?
Given a $C^*$-algebra $A$ and a state $\rho$ on $A$, let $\pi_\rho$ be the corresponding GNS representation on the Hilbert space $H_\rho$. I would like know when the image of $\pi_\rho$ is WOT-dense ...
4
votes
1answer
62 views
Smallness/ Rigidity of $\kappa(\mathcal{H})$ without using minimal projections?
Let $\mathcal{H}$ be a Hilbert space and $\kappa(\mathcal{H})$ the $C^*$-algebra of compact operators on $\mathcal{H}$. By smallness/ rigidity of $\kappa(\mathcal{H})$ I am referring to the following ...
4
votes
1answer
80 views
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 ...
4
votes
0answers
71 views
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 ...
4
votes
0answers
66 views
C* algebra of bounded Borel functions
Let $T\in B(H)$ is normal, and $B(\sigma(T))$ denote the $C^*$ algebra of all bounded Borel functions on $\sigma(T)$. Then is it true that $B(\sigma(T))$ is a closed $C^*$ algebra under the sup. norm ...
4
votes
0answers
113 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 ...
4
votes
0answers
153 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 $C(X)$ ...
4
votes
0answers
63 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, ...
4
votes
0answers
84 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. ...
4
votes
0answers
296 views
Double dual of the space $C[0,1]$
The second dual or double dual of the space of all continuous functions on $[0,1]$, $C[0,1]$ is von Neumann algebra. Can anyone help me identifying this space?
3
votes
2answers
425 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 ...
3
votes
3answers
122 views
Lower bound for $\|A-B\|$ when $\operatorname{rank}(A)\neq \operatorname{rank}(B)$, both $A$ and $B$ are idempotent
Let's first focus on $k$-by-$k$ matrices. We know that rank is a continuous function for idempotent matrices, so when we have, say, $\operatorname{rank}(A)>\operatorname{rank}(B)+1$, the two ...
3
votes
2answers
207 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 ...
3
votes
1answer
294 views
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 ...
