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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318 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
272 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
166 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
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358 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
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1answer
114 views

Can $xy$ and $yx$ lie in different connected components of the group of invertible elements of an algebra?

What is an example of a Banach or $C^{*}$ algebra $A$ which has two invertible elements $x, y$ such that $xy$ can not be connected to $yx$ in $G(A)$, the space of invertible elements of $A$. A ...
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161 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 $a\in\mathcal A$ ...
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228 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
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538 views

On the exponential form of a unitary matrix

A unitary matrix $U \in \large C_{n,n}$ can always be written in the exponential form: $U = e ^{iA}$ (1) where $A$ is Hermitian. My goal is to find the Hermitian matrix $A$, given the unitary ...
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0answers
156 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 ...
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93 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, ...
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4answers
579 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 ...
5
votes
1answer
267 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 ...
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votes
3answers
933 views

Sufficient condition for a *-homomorphism between C*-algebras being isometric

Let $\mathcal{A},\mathcal{B}$ be two unital C*-algebras and consider a *-homomorphism $\pi: \mathcal{A} \rightarrow \mathcal{B}$. I know that in general $\pi$ is contractive, i.e. $\vert\vert \pi(A) ...
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votes
2answers
178 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 $f_y(x)=B(x, y) \forall x \in E$. ...
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1answer
1k 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 ...
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154 views

Spectrum of normal elements in C*-algebras

Let $\mathcal{A}$ be a C*-algebra and $x \in \mathcal{A}$ a normal element. Can you show that $\left\{ \phi(x) : \phi \text{ is a state on } \mathcal{A} \right\}$ is the closed convex hull of the ...
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302 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
117 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 $b\in A$ ...
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220 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 ...
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300 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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1answer
104 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, ...
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147 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 ...
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293 views

Relation between noncommutative geometry and functional analysis

Recently I came across the subject of noncommutative geometry via my interest in functional analysis. My very little exposure to this subject gives me a sense that part of it is built on the theory of ...
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567 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 ...
5
votes
1answer
145 views

Construction of noncommutative torus

In short, how do we get the formula for the NC torus? I find the equations in many places (including here) but I still have no idea for how this comes from the torus. If my understanding is correct, ...
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votes
1answer
372 views

Physical interpretation of $q$-deformation

I am currently reading the paper Quantum Group Particles and Non-Archimedean Geometry by Volovich and Aref'eva. Here they discuss the difference between $q$-deformation and $\hslash$-deformation. In a ...
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1answer
42 views

Is $\mathbb{C}^n$ a C*-algebra?

Hi I am new in learning C*-algebra. I know that under the usual $\mathbb{C}^n$-norm, $\mathbb{C}^n$ is a Banach Space. However, what will be an intuitive multiplication on $\mathbb{C}^n$ so that the ...
5
votes
1answer
158 views

Cap product between K-Theory and K-Homology

In Exercise 9.8.9 of the book "Analytic K-Homology" by Higson and Roe one has to construct a cap product $K_p(A) \otimes K^q(A) \to K^{q-p}(A)$, if A is commutative. Is the commutativity ...
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204 views

Spectral measures

Let $E:\Sigma\to\mathcal{L}(\mathcal{H})$ be a spectral measure on the Borel $\sigma$-algebra $\Sigma$ of $\mathbb{C}$. Assume also that $E$ is compactly supported in the sense that ...
5
votes
1answer
77 views

Most natural equivalence between $C^*$-algebras

I have listen or read that, in the context of noncommutative geometry, Morita equivalence is a more natural equivalence for $C^*$-algebras than $*$-isomorphism. Can someone explain this sentence or ...
5
votes
1answer
79 views

Is there a cyclic vector for $-\frac{d^{2}}{dx^{2}}$ on $L^{2}[0,2\pi]$ with periodic conditions?

Let $\mathcal{H}=L^{2}[0,2\pi]$, and let $L=-\frac{d^{2}}{dx^{2}}$ on the domain $\mathcal{D}(L)$ consisting of twice absolutely continuous functions $f$ on $[0,2\pi]$ with $f''\in\mathcal{H}$ and ...
5
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1answer
80 views

Show this representation is irreducible and faithful

Let A be a prime $C^{*}$ algebra and $e= \alpha^{-1} c^{*}c$ for some $c \in A$ which satisfies $(c^{*}c)^{2}=\alpha c^{*}c$. Then $e^{2}=e=e^{*}$ and $eAe=\mathbb Ce$ so $eAe$ may be identified with ...
5
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1answer
204 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 ...
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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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Is the left regular representation of an algebra, always faithful?

Let $\mathcal{A}$ be a unital associative algebra with a countable basis $\mathcal{b}$ over $\mathbb{C}$. Let $H=l^2(b)$ be the Hilbert space generated by $\mathcal{b}$. Let $H_0 = \{v \in H \ \vert \ ...
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Conditional expectation onto maximal abelian subalgebras

If you take a von Neumann algebra $M$ and any its maximal abelian subalgebra (masa) $D$, then there is a norm-one projection from $M$ onto $D$ (conditional expectation). The same is true if you take ...
5
votes
1answer
260 views

Positive Operators: Definition?

Definitions Given an operator algebra $\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$ with $1\in\mathcal{A}$ Consider selfadjoint operators $A=A^*\in\mathcal{A}$. Define positive elements by: ...
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votes
1answer
358 views

application of c*algebras to PDEs

I am preparing an introductory talk about c* algebras and I'd like to motivate C*algebras or show an application of them and I'd prefer an application in the field of partial differential equations. ...
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38 views

freely generating elements in an algebra

Let $(\mathfrak{M}, \tau)$ be a W${}^{\ast}$-Algebra with (finite, normal, etc., whichever nice conditions one may find need for) tracial state. Elements $(a_{i})_{i\in I}\subset\mathfrak{M}$ shall be ...
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0answers
57 views

How can I show that given a norm one linear functional on $c_0$ that there is a unique extension to a norm one functional on $\ell_\infty$?

We are given that our Banach space is $c_0 \subset \ell_\infty(\mathbb{N})$ and there is a functional $y^* \in c_0^*$ such that $||y^*|| = 1$. We are guaranteed that this extends, via Hahn-Banach to a ...
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Generalisation of the trace cyclic theorem to partial traces.

The trace-cyclic theorem says that linear operators commute (or cycle) within the trace, that is $${\rm Tr}(XY) = {\rm Tr}(YX).$$ Now if $X$ and $Y$ operate on a product space $H_A \times H_B ...
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Masas in quotients

Let $A$ be a von Neumann algebra and let $B$ be a norm-closed ideal of $A$ (but not necessarily WOT-closed). What one has to assume about $A$ and $B$ to ensure that if $M\subset A$ is a maximal ...
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1answer
180 views

Minimal projections vs maximal left ideals

I've seen in some papers a statement (which is referred to a very old book of Dixmier in French which I have no access to / can't read anyway) saying that maximal left ideals of a (unital) C*-algebra ...
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127 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
1answer
118 views

Faithful Representations of C*-algebras

Can anyone give me an example of a represetation of the algebra $M_n(\mathbb{C})$ that is not faithul? If it's not possible, could you explain me why it is not?
4
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1answer
363 views

The set of all continuous functions on a locally compact Hausdorff space.

I am reading a book about C*-algebra. There is a example that i could not understand. Let $X$ be a locally compact Hausdorff space and $C_{0}(X)$ be the set of all continuous functions vanishing at ...
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3answers
179 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 ...
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469 views

Examples of not completely bounded maps

Let $\phi:\mathcal{A}\longrightarrow\mathcal{B}$ be a bounded map between $C^*$ algebras. $\phi$ is said to be completely bounded if the natural extension map \begin{eqnarray} ...
4
votes
2answers
190 views

Commutative unital Banach algebra with nilpotent elements

What would be a concrete example of a commutative unital Banach algebra that contains nilpotent elements?
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263 views

How generalize the bicommutant theorem?

Let $H$ be an infinite dimensional separable Hilbert space. Bicommutant theorem : Let $\mathcal{S}$ be $*$-subset of $B(H)$, then $\mathcal{S}''$ is the strong closure $\overline{\langle ...