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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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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101 views

The operator norm of complex matrices

Let $M_{n}$ be the algebra of $n\times n$ complex matrices. By identifying $M_{n}$ with $B(\mathbb{C^{n}})$, the set of all bounded linear maps from the n-dimensional Hilbert space $\mathbb{C^{n}}$ to ...
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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||= ...
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Norms arising from all representations of *-algebras

It is common that in order to obtain a $C^*$-algebra from a $^*$-algebra $A$ one defines a norm on $A$ by $$\|x\|=\sup\{\|\pi(x)\|\,|\,\pi\ \text{is a }^*\text{-representation of }A\}.$$ However, I ...
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Question about operators on Hilbert space

Let $\cal{H}$ be a Hilbert space, $P_1,P_2,\cdots,P_m$ a sequence of orthonormal projections such that $P_iP_j=0$ for $i\neq j$ and $P_1+P_2+\cdots+P_m=I$. Then $\|\sum^m_{k=1}P_kTP_k\|\leq\|T\|$ for ...
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98 views

An quasi-nilpotent operator restricted to a subspace is a nilpotent?

I am reading a paper about operator theory, there is a proposition I could not understand. Let $T\in L(X)$ be a quasi-nilpotent operator and $X_{1}$ be a non-zero finite-dimensional subspace of X, ...
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47 views

When does Gelfand Naimark Theorem Hold?

I was going through the proof of the Gelfand Naimark Theorem for the Unital Commutative Banach Algebras. In proving that each character has norm equal to $1$, we used the fact that $||e|| = 1$ where ...
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70 views

Kernel of an infinite dimensional operator

Can the Kernel of an infinite dimensional operator have $\dim=0$? I am thinking to the annihilation ($\hat E$) and creation ($\hat E^\dagger$) operators. Suppose, in fact, we have an infinite but ...
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248 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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73 views

number of generators of MASA

Let $\mathcal{H}$ be an infinite-dimensional Hilbert Space. Do the maximal abelian self-adjoint subalgebras of $\mathcal{B}(\mathcal{H})$ always have infinitely many generators as an algebra ? (The ...
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92 views

Left support of an operator on a Hilbert space

The left support $l(x)$ of an operator $x$ between Hilbert spaces $\mathbb{H}$ and $\mathbb{K}$ is defined as the smallest projection $e \in \mathfrak{B}(\mathbb{H})$ such that $ex=x$. The question ...
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Projections in group $C^*$-algebras

Let $G$ be an amenable, discrete and infinite group. Cosinder its group C*-algebra $C^*(G)$ canonically represented on $B(\ell_2(G))$ by the left-regular representation $x\mapsto \delta_x$. Take the ...
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260 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. $\to$ ...
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59 views

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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272 views

universal C* algebras

Is there a standard reference which has a discussion on universal $C^*$-algebras ? (definition, properties, examples, etc) Searching on the internet has led me to tidbits of information but I would ...
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170 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 : ...
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175 views

How to generate algebraic span of a set of matrices (how many multiplications?)

I've got a question about matrices and matrix algebras that offhand seems difficult, I'm wondering there is any sharp solution? Or perhaps it's known to not have any solution at all? Suppose you have ...
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79 views

Lorentz group and eigenvalues

For generators of the Lorentz group ($\hat {R}_{k}$ corresponds to the generators of 3-rotations, $\hat {L}_{k}$ corresponds to the generators of the boosts) we have the following algebra: $$ [\hat ...
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182 views

Does an irreducible operator generate an exact $C^{*}$-algebra?

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Definition : An operator $T \in B(H)$ is irreducible if $W^{*}(T)=B(H)$. Definition : A ...
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1answer
75 views

C* identity origins

In the context of $C^*$-algebras , why is the $C^*$-identity a "natural" one to choose ? ($||a^* a||=||a||^2$). Some books try to motivate this by noting that bounded operators on a Hilbert space ...
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1answer
190 views

Does an irreducible operator generate a nuclear $C^{*}$-algebra?

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Definition : An operator $T \in B(H)$ is irreducible (Halmos 1968) if its commutant $\{ T\}'$ ...
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405 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 ...
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Connection in the KK-Theory

I have some questions about the connection in the KK-Theory. 1)The definition is complicated, why? What is the motivation? 2)Does any relation bewteen the connection at here with the differential ...
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1answer
113 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} ...
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188 views

Hahn-Banach Theorem in the C*-algebra

What is the Hahn-Banach Theorem in the C*-algebra(or W*-algebra maybe)? If B is an nondense subalgebra of C*-algebra(or W*-algebra maybe), can we get an state f of A which is always zero at the ...
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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 ...
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Commutant of bounded linear operators on a Hilbert space

Given a Hilbert space $H$, denote by $\mathcal{A}=\mathcal{B}(H)$ the C*-algebra of bounded linear operators on $H$. Denote further by $$\mathcal{B}(H)' := \{A\in \mathcal{B}(H) : [A,B]=0 \;\forall ...
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Factorizing a saddle point operator

I have a coupled pde which, after semi-discretizing in time, results in the solution of a sequence of continuous saddle point problems $Lu^t=b$. Written explicitly, the problem looks like this: ...
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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 ...
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203 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 ...
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200 views

How to prove $[x,p]=i$ $\implies$ $[x,p^n]=inp^{n-1}$?

How to prove $[x,p]=i$ $\implies$ $[x,p^n]=inp^{n-1}$? I can do this using $p=i\frac{d}{dx}$, but my book hasn't introduced this yet so is there another proof without using this ? These are just ...
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Questions about $B(H)$ and $B(H)/K(H)$ as Banach space

I am trying to investigate the relation between Uniformly Convexity and existence of Schauder Basis for a Banach space. I read in a Handbook article that $B(H)$ (the algebra of all bounded operators ...
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494 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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1answer
209 views

Cyclic vectors of an irreducible representation of a C*-algebra

Let $\mathcal{A}$ be a C*-algebra and $(H,\pi)$ an irreducible representation of $\mathcal{A}$. I want to prove the statement: all $\xi \in H$ are cyclic or $\pi(\mathcal{A})=\{0\}$ and ...
2
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1answer
80 views

Sufficient condition for self-adjoint subset of bounded linear operators on a Hilbert space being irreducible

Let $H$ be a Hilbert space and denote as $B(H)$ the bounded linear operators on $H$. Let $M$ be a subset of $B(H)$, s.t. for $A \in M$, also $A^* \in M$. How can one show that if the commutant has ...
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154 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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56 views

Stable group algebras

Let $G$ be a discrete group and let $C^*_r(G)$ be its reduced group C*-algebra. Is there any group $G$ for which we have $C^*_r(G)\cong M_2(C^*_r(G))$? Or more generally, $C^*_r(G) \cong ...
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111 views

Property of Banach algebra with involution

Let $\mathcal{B}$ be a Banach algebra with involution *. Is it always true that $\forall A \in \mathcal{B}: \| A \|^2 \geq \| A^* A \| $? (motivation: I read a proof that bounded linear operators on ...
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1answer
137 views

Prove the approximate identity from the unitization

Suppose $A$ is a $C^*$-algebra without unit, $A^+$ is a unitization of $A$, treat $A$ in the $A^+$, if $\{x_n\}$ in $A$ converge (or monotonous converge) to $1$ in $A^+$, does $\{x_n\}$ must be the ...
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Relationship of two generalizations of the real/complex calculus

On the one hand, one has the various functional calculi from Operator Algebras. The continuous functional calculus for C* algebras, the bounded borel functional calculus for Von Neumann Algebras, the ...
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Proof that operator is an isometry

A linear operator $L$ between complex spaces with inner product $U$ and $V$ is an isometry, only if $\left < Lu_i, Lu_j \right > = \left < u_i, u_j \right >$ for all $u_j, u_i$ from a ...
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1answer
58 views

Proof that restriction of hermitian operator to its invariant subspace is also hermitian

Proof that restriction of hermitian operator to its invariant subspace is also hermitian What would be the most elegant way to prove this?
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715 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 ...
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167 views

On the use of nets when defining operator topologies

Let's consider the strong operator topology and the weak operator topology on bounded operators of a infinite-dimensional Hilbert space $H$. When they define these operator topologies, some authors ...
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GNS-triplets for states on the matrix space: generalization to the infinite-dimensional setting

In a previous exercise, I have proven that states $\omega$ on the $C^*$-algebra $M_n(\mathbb C)$ correspond to a unique density matrix $\rho$ by the relation $\omega(A) = \mathrm{Tr}(\rho A)$. I was ...
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$‎‎\sigma(x)‎$ ‎‎‎‎is ‎contained ‎in ‎the ‎imaginary ‎axis ‎of ‎the ‎complex ‎plane

$‎A$ ‎is a‎ ‎C*-algebra ‎and ‎‎$‎x‎\in A‎$ ‎satisfies ‎‎$‎x‎^*=-x‎$.‎I want to show that ‎‎$‎‎\sigma(x)‎$ ‎‎‎‎is ‎contained ‎in ‎the ‎imaginary ‎axis ‎of ‎the ‎complex ‎plane.How i prove it?
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286 views

Self-adjoint projections of a C*-algebra as complete lattices?

In Blackadar's Operator Algebras, there is the following remark after the proposition II.3.3.1 : The projections in a C*-algebra do not form a lattice in general In the answer of this question, ...
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1answer
79 views

Inner product on a von Neumann algebra

Let $M$ be a $\sigma$-finite von Neumann algebra (one which admits a faithful normal state) acting on a Hilbert space $H$. Denote its faithful normal state by $\omega$. We can define an inner product ...
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a trace class operator problem

Could someone help me with this Prove that If $A$ and $B$ are positive trace class operators on a Hilbert space, then so is $A^zB^{(1-z)}$ for a complex number $z$ such that $0 <Re(z)< 1$. An ...
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Arveson index of a completely positive map on matrix algebra

Can someone tell me what is Arveson index of a completely positive map. What I want is given a map \begin{eqnarray} \psi:\mathcal{B}(\mathbb{C}^m)&\longrightarrow&\mathcal{B}(\mathbb{C}^n)\\ ...