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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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
114 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 ...
2
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50 views

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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72 views

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
51 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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462 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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135 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 ...
3
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160 views

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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213 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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66 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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189 views

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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0answers
58 views

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)\\ ...
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77 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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55 views

$\ast$-homomorphism

Let $\phi: C(X,M_{4}(\mathbb{C})) \rightarrow C(Y,M_{8}(\mathbb{C})) $ be a $\ast$-homomorphism where $X$ and $Y$ are compact Hausdorff spaces. Let $M_{2}(\mathbb{C})$ be the C*-subalgebra of ...
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1answer
58 views

Subalgebras of certain C*-algebras

Let $A$ be a C*-subalgebra of $C(X, M_{n}(\mathbb{C}))$ where $X$ is a compact Hausdorff space, does it follow that $A$ is isomorphic to $C(Y, M_{m}(\mathbb{C}))$ for some $Y\subseteqq X$ and ...
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78 views

A $*$-homomorphism from the CAR algebra to $\mathfrak B(\mathcal H)$

Could a $*$-homomorphism $\pi:\text{CAR}\to\mathfrak B(\mathcal H)$ exist (with $\mathcal H$ separable) such that there is a compact and positive element $h\in\mathcal K$ commuting with the image of ...
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1answer
48 views

Spectrum of T in $B(\ell^2)$

Let $T:\ell^2 \to \ell^2$ be an operator on $\ell^2$ is defined as follows: $$T\{a_1,a_2,\dots\}=\{0,a_1,a_2,\dots\}$$ What is spectrum of T in $B(\ell^2)$?
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40 views

question about idempotent operators [duplicate]

Let $H$ be a Hilbert space and $\dim(H)=\infty$. If each $T\in B(H)$ is finite sum of idempotent operators? If each $T\in B(H)$ is infinite sum of idempotent operators?
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Question about derivation in Jordan algebra

Let $(G,\circ)$ be a Jordan algebra, then $\sigma:G\to G$ given by $$c\mapsto a\circ(b\circ c)-b\circ(a\circ c),\quad\forall c\in G,$$ is a derivation, where $a$ and $b$ are two fixed elements of $G$. ...
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1answer
572 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 ...
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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 ...
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1answer
109 views

K-theory, $K_{0}$ of algebra of compact operators

I don't understand how to define the trace of a matrix with values in operators. This occurred in the following situation: Suppose that $H$ is an Hilbert space and $K$ is the algebra of compact ...
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1answer
82 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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1answer
66 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 ...
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1answer
174 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 ...
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1answer
195 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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1answer
112 views

Quotients of the maximal tensor product

Let $A$ and $B$ be C*-algebras and let $\gamma$ be any C*-norm on the algebraic tensor product $A\odot B$. Why is $A\otimes_\gamma B$ a quotient of $A\otimes_{{\rm max}}B$, where $\otimes_{{\rm max}}$ ...
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99 views

Bounded Operator with Closed Range

I've read Martin Argerami's answer to this question. On the first line he claims that the range of $T$ is closed. Can somebody explain me why that's the case? For me it is not necessarily closed.
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137 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 ...
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118 views

From positive definite function to Følner sequence -— a question on amenability and nuclearity

We know that amenability of countable discrete group $\Gamma$ has many equivalent characterizations. In particular, there are two: a) there is a sequence of finitely supported positive definite ...
3
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1answer
59 views

Block Matrices of Operators

I'm trying to prove the following: Consider the vector space of matrices of size $n\times n$ whose entries in $\mathcal B(H)$. Denote this vector space by $M_{n,n}(\mathcal{B(H)})$. We can define ...
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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 $b\in A$ ...
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1answer
94 views

On the Spectral Theorem

Let $H$ be a Hilbert space, $T\in B(H)$ be normal and $E$ its spectral measure. a- Let $\delta >0$ , and let $M_{\delta}$ = $\left\{\lambda\in \sigma(T): |\lambda|\geq \delta\right\}$. ...
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1answer
164 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 ...
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1answer
74 views

A question concering nuclearity

B. Blackadar in his book Operator algebras: Theory of C${}^\ast$-Algebras and Von Neumann Algebras defines a C*-algebra $A$ to be nuclear if for every C*-algebra $B$ the algebraic tensor product ...
3
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1answer
86 views

Algebra (Not *)-Isomorphisms of von Neumann algebras

Let $A$ and $B$ be any two infinite-dimensional von Neumann algebras, they are operator algebras with operator composition as the multiplication and as infinite dimensional vector spaces they're ...
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1answer
210 views

Point spectrum in Hilbert spaces

Let $H$ be a Hilbert space and and $T\in B(H)$ be normal and $\sigma_p(T)$ be the point spectrum of $T$ (i.e the set of all eigenvalues of T) and let $E$ denote the spectral measure. I'm trying to ...
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2answers
205 views

Unitary operator in von Neumann algebra

Let $R\subseteq B(H)$ be a von Neumann algebra, and $U\in R$ be unitary. Prove that there is a self adjoint operator $A\in R$ such that $||A||\leq \pi$, and $U=\exp(iA)$ . Any idea how to start! Thank ...
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1answer
67 views

What is $\sigma_{\mathfrak{M}_{p}}(p\cdot a\cdot p)$ when $p=\chi_{S}(a)$ for closed $S\subseteq \sigma(a)$?

Let $\mathfrak{M}$ be a von-Neumann algebra and $a\in\mathfrak{M}$ a self-adjoint element. Let $S\subseteq\sigma(a)$ be a non-empty, closed subset. Let $p=\chi_{S}(a)$, which is a non-zero, ...
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1answer
67 views

differential equation of the square root of a matrix

If the differential equation governing the time dependent matrix $M(t)$ is $\frac{dM(t)}{dt}=A.M(t).B$ or $\frac{dM(t)}{dt}=A.M(t)+M(t).A$ where $A$ and $B$ are constant matrices, what is the ...
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190 views

Proving properties of exponential map on a Banach algebra

$$\exp(a) := \sum\frac {a^k}{k!}$$ Can you help me prove that: $\exp$ is well defined (i.e. converges for all $a$ in $A$) $\exp$ is continuous $\exp(A)$ is a subset of $A_0$ (where $A_0$ is the ...
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44 views

Are decomposable maps completely bounded?

By the word decomposable I mean a positive map $\phi:\mathcal{B(H)}\rightarrow \mathcal{B(K)}$; $\mathcal{H,K~}$ are some Hilbert spaces and $\phi=\psi_1+T\circ \psi_2$ where $T$ is the transpose ...
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35 views

A map that is $(n-1)$-positive but not $n$-positive

Let $\phi : M_n(\mathbb{C})\to M_m(\mathbb{C})$ be a linear map. $\phi$ is called $k$-positive if the map $\phi^{(k)} : M_{kn}(\mathbb{C}) \to M_{km}(\mathbb{C})$, defined by evaluating $\phi$ ...
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118 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 ...
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1answer
94 views

Is there any operator which its spectrum corresponding to a compact set?

we know that for each operator $T$ the spectrum $\sigma(T)$ is compact. Is the converse true I mean if we have a compact set $K\neq\emptyset$, is there any operator $T$ such that $\sigma(T)=K$? I am ...
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1answer
45 views

Are self-adjoints elements norming?

Let $\mathsf A$ be a C*-algebra and let $\mathsf{A}^*$ be its dual space. Is it true that for $f\in \mathsf A^*$ we have $$\|f\|=\sup\{|f(x)|\colon\; \|x\|=1\mbox{ and }x^* = x\}?$$
2
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1answer
73 views

ideals in $C^*$ algebra

Let $A$ be a $C^*$ algebra and $I$ be a closed ideal in $A$. Prove that for all $a\in A$, $a\in I$ iff $a^*a\in I$. I want to prove that if $a^*a\in I$, then $a\in I$, and I know the following fact ...
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1answer
80 views

strictly positive in unital $C*$ algebra

Let $A$ be a unital $C*$ algebra. $a\in A_+$ is called strictly positive if $\overline{aAa}=A$. Prove that $a\in A_+$ is strictly positive iff $a$ is invertable. I proved one direction : if $a$ is ...
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36 views

Finding Strictly Positive Elements [duplicate]

I need to find the set of strictly positive elements in the $C^*$-algebra $C_0(\Omega)$ where $\Omega$ is a locally compact Hausdorff space. Clearly, the set will be contained in $ \{ f \in ...