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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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 ...
17
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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 ...
17
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1answer
316 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 ...
14
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1answer
508 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
339 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
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2answers
711 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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1answer
421 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
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3answers
512 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
203 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
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2answers
617 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 ...
9
votes
2answers
217 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
315 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
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1answer
178 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
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1answer
310 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
1answer
560 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
2answers
190 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
378 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
120 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
114 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
392 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
199 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
616 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
48 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
151 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
123 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
359 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
1answer
303 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
328 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
122 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
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1answer
581 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
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1answer
257 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
208 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
248 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
110 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
284 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
144 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
187 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
314 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 $C(X)$ ...
5
votes
1answer
530 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
183 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
161 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
136 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 ...
5
votes
1answer
766 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
216 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
2answers
193 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 ...
5
votes
1answer
286 views

A theorem about operator theory

Define $$\operatorname{Ref}\mathcal{S}=\{T\in B(\mathcal{H}):Th\in[\mathcal{S}h], \forall h \in \mathcal{H}\},$$where $\mathcal{H}$ is a Hilbert space and $\mathcal{S}$ is a linear manifold of ...
5
votes
1answer
90 views

Definite states on C*-algebras

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
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1answer
83 views

Question about Hahn-Banach theorem

Let $(X,\|\cdot\|_1)$ and $(Y,\|\cdot\|_2)$ be normed spaces, and $X\subset Y$. If each $f\in (X,\|\cdot\|_1)^\ast$ extends to a bounded linear functional in $(Y,\|\cdot\|_2)^\ast$ with same norm, ...
5
votes
1answer
179 views

Nonamenable subgroups of the unitary group of the hyperfinite II_1 factor

The hyperfinite $II_1$ factor arises as the group von Neumann algebra of any infinite amenable group such that every conjugacy class but that of the identity has infinite cardinality. The unitary ...
5
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2answers
441 views

Regarding Ladder Operators and Quantum Harmonic Oscillators

When dealing with the Quantum Harmonic Oscillator Operator $H=-\frac{d^{2}}{dx^{2}}+x^{2}$, there is the approach of using the Ladder Operator: Suppose that are two operators $L^{+}$ and $L^{-}$ and ...