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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Can someone help me to give some hints? Left Hilbert-$C_0(T,K(H))$ module $C_0(T,H)$

I tried to prove example 3.4 from the book Morita Equivalence and Continuous-Trace C$^*$-Algebras by Iain Raeburn and Dana P. Williams, but I get uneasy with notations and ideas. Let me restate my ...
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73 views

What is an algebra and what is it's representation?

Heyho, i've kind of got an understanding problem what exactely an algebra and especially it's representation is. In my case it is said, that the relation $R_{12}(u-v) (L(u) \otimes \mathrm{I}) \; ...
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93 views

What, and how can, topological invariants can be computed from a space's algebra of functions?

The Gelfrand duality says that the category of locally compact Hausdorff spaces (with proper continuous functions) is equivalent to the category of commutative $C^*$ algebras (with proper ...
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1answer
55 views

Power series expansion of an Operator.

I've been reading a paper called "Separation of variables for the quantum $Sl(2,R)$ spin chain" in which the author at one point does a power series expansion I do not understand. The problem is this ...
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29 views

Clarification on Wolfram Mathworld's explanation of the connection between Gelfand Transform and Fourier Transform

http://mathworld.wolfram.com/GelfandTransform.html In the definition, what does $x$, $\hat x(\phi)$, and $\phi$ represent exactly if we were to consider definition of the Fourier transform? Can ...
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75 views

Is there a $C^{*}$ algebra with these properties

Is there a unital C* algebra A which is NOT simple but satisfies the following two conditions: 1)A has trivial center 2)A has a faithful trace such that every zero trace element lies in the closure ...
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25 views

Ordering: Normalized

Given a Hilbert space $1\in\mathcal{A}$. Denote selfadjoints: $$\mathcal{S}:=\{A\in\mathcal{A}:A=A^*\}$$ Introduce an order: $$A\leq A':\ \ \sigma(A'-A)\geq0$$ Regard a projection: $$P\neq0:\quad ...
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28 views

Weak (operator) null sequence is bounded and pointwise convergent to zero

I was reading Diestel book (Absolutely Summing Operators) and it says: "(...) let $(f_n)$ be any weak null sequence in $\mathcal{C}(K)$. Then $(f_n)$ is bounded and converges pointwise to zero." I ...
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1answer
43 views

Operators whose spectrum has a finite number of connected component

Assume that $H$ is a separable Hilbert space. Let $Q$ be the set of all operators$T \in B(H)$ such that the spectrum of $T$ has a finite number of connected component. Is $Q$ a subvector space ...
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18 views

Derivation into dense ideal of Banach algebras

Let $A$ be a Banach algebra and $I$ be an ideal of $A$. A derivation $D:A\to I$ is a linear bounded map, with the following property: $$D(ab)=aD(b)+D(a)b,\qquad a,b\in A.$$ Suppose that $I$ is dense ...
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37 views

A question about the weak operator topology of $B(H)$

Let $H$ be a Hilbert space and $V$ a dense subspace. Let $A$ be a $*$-subalgebra of $B(H)$. The weak closure of $A$ is, by definition, the space of all $u\in B(H)$ satisfying: For every ...
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27 views

Normal Operators: Superalgebra (II)

Problem highlighted at the end! Application Reduction to only one operator!! Reference This builds up on: Superalgebra (I) Convention All operators possibly unbounded!! Structures Given a ...
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30 views

Quasi ideal sequence in $B(H)$

According to comments by Hamza I revise the question. Let $H$ be an infinite dimensional separable Hilbert space. Is there an increasing sequence of subvector spaces $V_{1} \subsetneq V_{2} ...
2
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22 views

Distributivity of projective tensor product over direct sum

Let $I$ is a non-empty set and $\{A_i\}_{i\in I}$ is a family of Banach algebras and $B$ is a Banach algebra. Define $$\ell^1-\oplus_{i\in I}A_i=\{a=\{a_i\}_{i\in I}: \|a\|_1=\sum_{i\in ...
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1answer
25 views

composition and strong limits of completely positive maps is completely positive

I have two claims about completely positive maps. Let $X$, $Y$, $Z$ be $C^\ast$-algebras. 1) Let $f:X\to Y$ and $g:Y\to Z$ be completely positive maps. I want to know, why $g\circ f$ is completely ...
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1answer
31 views

the $C^\ast$-algebra $M_n(A)$, understanding the $C^\ast$-norm on $M_n(A)$

Let $A$ be a $C^\ast$-algebra. I want to understand $M_n(A)$, the vector space of $n\times n$-matrices with entries in $A$, as a $C^\ast$-algebra. On $M_n(A)$ you can define an involution ...
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1answer
40 views

The inverse limit of C$^*$-algebras and whether it commutes with taking the minimal tensor product

Suppose we are given a C$^*$-algebra $A$ and a family of C$^*$-ideals $\mathfrak{I}$ that is upwards directed when ordered by reverse inclusion (i.e. for any $I_1,I_2\in\mathfrak{I}$ there exists a ...
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18 views

Gaussian unitary dilation of Gaussian channels

I am starting with a few definitions. All these are standard and can be accessed from some quantum information or quantum physics books, for instance the books by Holevo or Parthasarathy. The question ...
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23 views

Semiprimitivity of second dual of semiprime Banach algebras

Let $A$ be a Banach algebra. Then $A^*$ is right Banach $A$-module with product $\langle b,f.a\rangle=\langle ab,f\rangle$ for every $a,b\in A, f\in A^*$. Define $\langle a,F*f\rangle=\langle ...
3
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23 views

Weakly compact operator with different domains

Let $A$ be a Banach algebra. Suppose that $e\in A$ such that $e^2=e$ and $eAe$ is division algebra(i.e., $eAe$ is unital and every element of $eAe$ has inverse in $eAe$). Define $T_e:A\to A$ with ...
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2answers
48 views

Projections: Orthogonality

Given a unital C*-algebra $1\in\mathcal{A}$. Consider projections: $$P^2=P=P^*\quad P'^2=P'=P'^*$$ Order them by: $$P\perp P':\iff\sigma(\Sigma P)\leq1\quad(\Sigma P:=P+P')$$ Then equivalently: ...
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29 views

Meaning of kernel

If something is the 'kernel' of a transformation, say $K(x,x')$, does it mean I should take the integral $$\int K(x,x') f(x')dx'$$ There are many different meanings of kernel and I did not see their ...
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Uniqueness of c.p.c. order zero extensions

I have a question about a passage in the proof (on page 316) of proposition 3.2 in this paper http://wwwmath.uni-muenster.de/42/fileadmin/Einrichtungen/mjm/vol_2/mjm_vol_2_14.pdf. My question is ...
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25 views

Relation between tracial states on von Neumann algebras and their GNS representations

Let $M$ be a von Neumann algebra acting on a Hilbert space $H$, and let $\tau$ be a faithful tracial state on $M$. What is the relation between the GNS representation of $(M,\tau)$ and the original ...
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35 views

Partial Isometries: Positivity

Given a unital C*-algebra $1\in\mathcal{A}$. Then implication holds: $$J\in\mathcal{A}:\quad JJ^*J=J\implies\sigma(J)\geq0$$ How can I check this? (Operator-algebraically?)
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Weakly compact left multipliers

This is Exercise 3(a) on p. 157 in Takesaki's Operator algebras. Let $A$ be a C*-algebra. Then each opeator $T_a\colon A\to A$ given by $T_ax = ax$ ($a\in A$) is weakly compact if and only if $A$ is ...
2
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1answer
37 views

Flip automorphism for a $II_1$ factor is not inner

It is known that for a $II_1$ factor $M$, the flip automorphism defined on $M \overline{\otimes} M$ by $a \otimes b \mapsto b \otimes a$ is not inner. A proof can be found on Vol. IV of the books by ...
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54 views

Some questions about Cuntz’s proof of the $ K_{1} $-injectivity of purely infinite simple unital $ C^{*} $-algebras

I have some questions about Joachim Cuntz’s proof of the $ K_{1} $-injectivity of purely infinite simple unital $ C^{*} $-algebras, which is found in this paper. For this post, let us adopt the ...
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40 views

tensor-operator- schatten norm

let $H_1$ and $H_2$ are hilbert spaces with dimention $n$ and $m.$ ‎assume $$\varphi:\mathcal{B}(H_1)\hat{\otimes}\mathcal{B}(H_2)\to\mathcal{B}(H_1\hat{\otimes}{H_2)},$$ which is defind as ...
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Ordering: Identity

Given a unital C*-algebra $1\in\mathcal{A}$. Denote selfadjoints: $$\mathcal{S}(\mathcal{H}):=\{A\in\mathcal{B}(\mathcal{H}):A=A^*\}$$ Introduce an order: $$A\leq A':\iff\sigma(A'-A)\geq0$$ ...
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1answer
53 views

Projections: Spectrum

Given a unital C*-algebra $1\in\mathcal{A}$. For projection one has: $$P^2=P=P^*\iff\sigma(P)\subseteq\{0,1\}\quad(P=P^*)$$ And all cases can appear: ...
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81 views

Projections: Ordering

Given a unital C*-algebra $1\in\mathcal{A}$. Consider projections: $$P^2=P=P^*\quad P'^2=P'=P'^*$$ Order them by: $$P\leq P':\iff\sigma(\Delta P)\geq0\quad(\Delta P:=P'-P)$$ Then equivalently: ...
2
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0answers
11 views

Finding minimal projections in subalgebra generated by a given set

Consider the set of complex matrices $\mathbb{C}^{n\times n}$ for some set. Suppose we have a set $\{A_1,\ldots, A_n\}$ of Hermitian matrices. We want to find minimal projections in the subalgebra ...
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78 views

If a C*-algebra $A=\overline{\bigcup S}$, where $S$ is a class of prime C*-subalgebras, then $A$ is prime.

This is question 5.6 of Murphy's C$^*$-Algebras and Operator Theory: Let $S$ be a set of C*-subalgebras of a C*-algebra $A$ that is upwards-directed, that is, if $B,C\in S$, then there exists ...
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closed range bounded linear operators

Let $CL(X,Y)$ be the set of all closed range bounded linear operators from Banach space $X$ to Banach space $Y$. Is $CL(X,Y)$ an open set of $B(X,Y)$?
3
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1answer
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Strict positiveness on a C*-algebra given by generators and relations.

Let $A$ be a C*-algebra with generators $a_1,a_2,\ldots,a_n$ and some (non-important) relations (the relations imply that $\|a_i\|\leq 1$, so that $A$ exists). Among the given relations we have that ...
3
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1answer
66 views

Is the supremum norm the only $ C^{*} $-norm on $ {C_{c}}(X) $, equipped with the usual pointwise operations?

Let $ X $ be a locally compact Hausdorff space. Then $ {C_{c}}(X) $ is a commutative $ * $-algebra with respect to addition, multiplication, scalar multiplication and conjugation (all pointwise ...
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1answer
49 views

When the Multiplier algebra of a Banach algebra is exactly equal to the operator algebra?

Let A be a Banach algebra. B(A) and M(A) be the operator algebra and the multiplier algebra of A, respectively. When we have M(A)=B(A)?
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1answer
15 views

Can a sequence of von Neumann algebras determine a maximal directed set of subalgebras?

Can a von Neumann algebra $A$ have an infinite sequence $A_0 \subset A_1 \subset A_2 \subset ...$ of sub-vN-algebras such that every other sub-vN-algebra $B \subseteq A$ satisfies, for some $n \geq 0$ ...
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26 views

Questions about stable rank of inductive limit of $C^\ast$-algebras

Let $A$ be an inductive limit of $\{A_n\}$ which are stable rank one. In Huaxin Lin's book An introduction to the classification of amenable $C^\ast$-algebra. The author assume that $\{A_n\}$ and $A$ ...
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1answer
19 views

Matrix factorization inequality

How does one show that the following matrix factorization inequality holds in $M_{n} (\mathcal{A})^{+}$, $$(a_{i}^{*}a^{*}aa_{j}) \leq ||a^{*}a|| \cdot (a_{i}^{*}a_{j})$$ Notation. Let $M_{n} ...
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33 views

Centralizer of $C^*$ algebra

Let $\phi: A \to B$ be a surjective $∗$-homomorphism of a separable $C^*$algebra. If $L: A \to A$ is a left centralizer then the formula $\phi(L)(\phi(a)) = \phi(L(a))$ defines a left centralizer for ...
3
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58 views

Projective (or inverse) limit of C*-algebras

(I think that the term "inverse limit" is used when the index set is directed) To begin with, I'd like to know if projective limits of C*-algebras (in the category of C*-algebras) always exist, and ...
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Two-sided closed ideals of $C(X,M_2(\mathbb C))$

Let $X$ be compact and Hausdorff space. I know all closed ideals of $C(X)$. I want to substitute $\mathbb C$ by $M_2(\mathbb C)$. What can we say about two-sided closed ideals of $C(X,M_2(\mathbb ...
3
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2answers
58 views

Let $A$ be a $C^*$-algebra, $a \in A$ self adjoint

Question:Let $A$ be a $C^*$-algebra, $a \in A$ self adjoint. Suppose that the spectrum $\sigma(a)$ is an infinite set. Show that $A$ is infinite-dimensional. How can i prove it? I guess: Let $A$ be ...
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39 views

Question about trace class operators

Let $\cal{H}$ be a Hilbert space, $T$ a bounded linear operator on $\cal{H}$, $S$ a trace class operator, then can one verify that $$|Tr(TS)|\leq\|T\|\cdot|Tr(S)|?$$
3
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57 views

Pureness of Vector States

How does one show that irreducibility is equivalent to a vector state being pure? In what follows I will fill in the details of the question: Let $\mathcal{H}$ denote a Hilbert space and let ...
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1answer
47 views

Noncommutative torus with $\theta = 0$.

According to Wikipedia, one can construct noncommutative tori as follows: Letting $\theta \in \mathbb{R}$ be a parameter, consider the hilbert space $H = L^2(\mathbb{T})$, where $\mathbb{T}$ is the ...
4
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49 views

Prove that the map $f: \alpha I + \beta A^*A \mapsto \alpha + \beta ||A||^2 $ is unital

Fix $A \in {\mathcal{A}}$, where $({\mathcal{A}},||\cdot||,*)$ is a $C^*$-algebra with unit $I$. Prove that the linear form $$ f(D_{\alpha,\beta}) = \alpha + \beta ||A||^2,~~D_{\alpha,\beta} \in ...
4
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1answer
42 views

Questions about Stinespring's theorem for completely positive maps

I have a question about Stinespring's Theorem: Let $A$ be a $C^*$-algebra, $H$ be a complex Hilbert space and $L(H)$ the set of bounded linear operators on H. Let $\Phi: A\to L(H)$ be a completely ...