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 ...

learn more… | top users | synonyms (1)

0
votes
1answer
37 views

How to prove that $A B A^* \leq \|B\| A A^*$ for operators A,B?

Let $A$, $B$ bounded operators on a Hilbert space $H$. Further let $B$ be self-adjoint. Then we have that $A B A^* \leq \|B\| A A^*$. I wanted to ask how to prove this inequality or where I can find ...
0
votes
1answer
60 views

Closed graph theorem question?

Let $H$ be a Hilbert space. Let $A:\operatorname{dom}A\to H$ has a closed graph, where $\operatorname{dom}A$ is dense in $H$. Let $S\subseteq \operatorname{dom}A$ be dense. Is it true $A_{|S}$ has a ...
2
votes
1answer
56 views

A funny way of expressing the identity operator

I have encountered the following trick that people in C*-algebra use; but frankly I don't understand why really this is true. Let $A$ be a unital C*-algebra acting non-degeneratly on a Hilbert space ...
2
votes
2answers
35 views

An inequality for completely positive maps.

Let $f\colon A\to B$ be a contractive completely positive, ${}^*$-preserving map between C*-algebras and take $a\in A$. How one can prove that $$0\leqslant f(a)f(a^*)\leqslant f(aa^*)?$$ Some authors ...
4
votes
3answers
143 views

Proof: $C(X×Y)=C(X)⊗C(Y)$

Where I can find the proof of the following theorem: Let $X$ and $Y$ be compact Hausdorff spaces, $C(X)$ and $C(Y)$ the space of continuous functions on $X$ and $Y$ respectively, then we have ...
1
vote
1answer
147 views

Positive elements of a $C^*$ (MURPHY, ex 2-2).

I'm studying "MURPHY, $C^*$-Algebras and Operator Theory" thoroughly and got stuck in the following exercise: Exercise 2, chapter 2. Let $A$ be a unital $C^*$-algebra. (a) If $a,b$ are positive ...
1
vote
1answer
26 views

Question about equivalent representations of von Neumann algebras in Kadison's book

When I read Kadison's book Fundamentals of The Theory of Operator Algebras, I meat a qeustion on page 460, at line 14, I can not understand that $A\mapsto AE'$ and $A\mapsto \Phi_n(A)F'$ are unitarily ...
0
votes
0answers
40 views

A conjugation of an operator, which commutes with all permutations, still commutes with all permutations

Assume $v:H^{\otimes m}\to H^{\otimes m}$ is a linear operator on the $n^{\text{th}}$ tensor power of a vector space $H$. For each permutation $p$ on the $m$-element set define the linear operator ...
4
votes
1answer
61 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 ...
1
vote
0answers
40 views

Rotation semigroup and identity element.

Let $\Gamma =\{z\in\mathbb{C}:|z|=1\}$, and $X=C(\Gamma)$. The rotation semigroup $\{T(t)\}_{t\geq 0}$, is defined as $$T(t)f(z)=f(\mathrm{e}^{it}z),\quad f\in X.$$ $Z\in X$, s.t. for all ...
1
vote
1answer
63 views

Cross Product Algebras references

Can someone give some references to introductory books or online notes about group algebras and cross-product algebras ? I've already searched on Google (but only for some online notes). The purpose ...
0
votes
1answer
35 views

Normal element of $C^* $ algebra whose spectrum lies in a certain direction in the complex plane

A is unital $C^*$ algebra and $a \in A$. I need to prove that set of elements {$1,a,a^*$} is linearly dependent if and only if $a$ is normal element and $\sigma(a)$ lies in direction in the complex ...
3
votes
1answer
63 views

Question about the type decomposition of von Neumann algebras, Blackadar's notes.

this is a little bit of a dumb question, please be nice, I had some doubts about the type decomposition of von Neumann algebras. I was reading Bruce Blackadar's "Operator algebras. Theory of ...
2
votes
1answer
203 views

Formal series expansion of differential operator (d/dx + f(x))^n

My original problem was to find the "coefficient" functions $\varphi_{k,n}(x)$ in $$ (\partial_x + f(x))^np(x) = \sum_{k=0}^n\varphi_{k,n}(x)\partial_x^kp(x). $$ (i.e. find the coefficients ...
2
votes
1answer
57 views

Approximate point spectrum and left topological zero divisors

Recall that a left topological zero divisor in a Banach algebra $A$ is an element $a\in A$ such that there exists a sequence of unit vectors $(a_{n})$ in $A$ with $\lim_{n\rightarrow\infty}aa_{n}=0$. ...
7
votes
1answer
135 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 ...
3
votes
1answer
41 views

A question on a sequence in a Banach algebra [duplicate]

If $\{u_{k}\}_{k=1}^{\infty}$ is a sequence in an Banach algebra (and more specifically, in the set of all the bounded linear operators of a Banach space $X$). If ...
3
votes
1answer
124 views

A question on the spectral projection

I am reading a paper about spectral theory. And I meet with some problems. An operator $K\in L(X)$ is said to be algebraic if there exists a non-trivial complex polynomial $h$ such that $h(K)=0$. By ...
2
votes
1answer
44 views

Exponential map in $C^{*}$-algebra and unitary invariance

Let $A$ be a unital $C^{*}$-algebra. Let $X$ be a closed vector subspace of $A$ which is unitarily invariant in the sense that $uXu^{*}\subseteq X$ for all unitaries $u$ of $A$. I want to show that ...
1
vote
0answers
37 views

Spectral radius as the inf of norms of conjugates

I need help with the following problem: Let $A$ be a unital $C^{*}$-algebra. (a) If $r(a)<1$ and $b=(\sum_{n=0}^{\infty}a^{*n}a^{n})^{1/2}$, show that $b\geq 1$ and $||bab^{-1}||<1$. (b) ...
6
votes
1answer
183 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 ...
2
votes
0answers
127 views

Conditional expectation on the space of bounded linear operators

In the paper from the link http://arxiv.org/pdf/0906.0139.pdf the author uses a diagonal conditional expectation. We take a seperable Hilbert space $H$ and fix an orthonormal basis $(e_n)_{n \in ...
3
votes
1answer
53 views

A simple question about completely positive linear maps

Let $A$ be the C*-algebra and $M_{n}(A)$ be the C*-algebra of $n\times n$ matrices with entries in $A$. We use $(a_{ij})$ to denote the element of $M_{n}(A)$. My question is: For every $a\in A$, ...
0
votes
1answer
65 views

Generator for $C_{0}(\Omega)$

Let $\Omega$ be a locally compact Hausdorff space, and suppose that the $C^{*}$-algebra $C_{0}(\Omega)$ is generated by a sequence of projections $(p_{n})_{n=1}^{\infty}$. Show that the hermitian ...
0
votes
1answer
96 views

Spectrum of a product

Let $A$ be a unital $C^{*}$-algebra. I am trying to show that if $a,b\in A$ are positive elements, then the spectrum of $ab$ is contained in the positive real numbers. I know that in the commutative ...
0
votes
1answer
55 views

Example of Hilbert space operator that is not a product of unitary and positive

If $A$ is a unital $C^{*}$-algebra, and $a\in A$ is invertible, then $a=u|a|$ where $u$ is unitary and $|a|=(a^{*}a)^{1/2}$ is positive. I am looking for an example of a bounded linear operator on ...
2
votes
1answer
113 views

When an invertible element in a $C^{*}$-algebra is unitary

I am trying to show that if $a$ is an invertible element of a unital $C^{*}$-algebra, and $||a||=||a^{-1}||=1$, then $a$ is unitary. I can do this if I think of $a$ as a Hilbert space operator using ...
3
votes
1answer
125 views

Question about projections on Hilbert space

Let $P_i$ be projections from a Hilbert space $\cal{H}$ to its closed subspace $\cal{H}_i$, $i=1,2,\cdots,n$, such that $\sum^n_{i=1} P_i$ is also a projection. And let $P$ be a projection from ...
2
votes
1answer
102 views

An exercise on C*-algebra

A representation $\pi$: $A\rightarrow B(H)$ is said to be irreducible if $\pi(A)$ has no non-trivial invariant subspace. A C*-algebra $A$ is said to be liminal if $\pi(A)=K(H_{\pi})$ for every ...
2
votes
1answer
77 views

How do we show that prime C* algebras have trivial center

A prime C* algebra is a C* algebra with the property that the product of any two of its non zero ideals is non zero. The claim is that it has trivial center, i.e., the only central elements are ...
1
vote
1answer
137 views

Proving the inclusion map induces isomorphism on $K$-theory

Let $M$ be a $C^\ast$-algebra, $A, B$ be closed, two-sided ideals of $M$ such that $A+B=M$. Define $T=\{f\in C([0, 1], M):f(0) \in A, f(1) \in B\}$. Why is that the inclusion map of $C([0, 1], A\cap ...
0
votes
1answer
62 views

A question about tensor product

If $A$ is an algebra, $M_{n}(A)$ denotes the algebra of all $n\times n$ matrices with entries in $A$. The operations are defined just as for scalar matrices. If $A$ is a *-algebra, so is $M_{n}(A)$, ...
1
vote
1answer
90 views

A question about projections in a von Neumann algebra

Let $\cal{R}$ be a von Neumann algebra $P\in\cal{R}$ a projection, $G$ a subprojection of $P$. Then the central carrier of $G+(I-C_G)E$ is $C_E$. Note: The central carrier of a projection $P$ is ...
1
vote
1answer
69 views

A question about range projection in von Neumann algebra.

I am reading a book about C*-algebra. And I meet with a problem. Recall the range projection of an operator $a\in B(H)$ is the projection on the closure of $\{a(\eta):\eta\in H\}$(Here, $H$ is a ...
2
votes
2answers
176 views

Problem with spectral theorem and spectral measure.

There is a passage in a book that is not very clear to me: A is a C*Algebra and $a$ is selfadjoint. Then "Indeed identifying A with an algebra of operators on a Hilbert space $\mathcal{H}$, by the ...
4
votes
1answer
104 views

MASAs of C* algebras

While studying the $C^*$-algebraic formulation of the recently solved Kadison-Singer problem, I was wondering about maximal abelian subalgebras: Let $\mathcal{A}$ be a unital C* algebra. There seems ...
2
votes
1answer
48 views

Give a counterexample that $A$, $B$ are similar matrices in $M_{n\times n}(\mathbb{C})$ but $PAP^{-1}\neq B$ for any $P\in GL_{n}(\mathbb{R})$.

Give a counterexample that $A$, $B$ are similar matrices in $M_{n\times n}(\mathbb{C})$ but $PAP^{-1}\neq B$ for any $P\in GL_{n}(\mathbb{R})$. How to construct this example? I have obtained that of ...
-1
votes
1answer
67 views

Why do closed ideals of C*-algebras have approximate units

In the Blackadar's book Operator algebras: theory of $C^\ast$-algebras and von Neumann algebras, on p. 103, there is a statement "approximate units for $J$", here $J$ is an ideal of a C*-algebra $A$, ...
1
vote
1answer
31 views

A simple question about the dimension of subspace.

I have a simple question: Let $A$, $B$ be closed subspaces of banach space $X$ and $B\subseteq A$, if $\dim A/B<\infty$ and $\dim B<\infty$, then $\dim A<\infty$? Why?
5
votes
0answers
52 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 ...
0
votes
1answer
91 views

A simple question in functional analysis

A classical result, in functional analysis, says that if $T\in B(X)$, the function: $\lambda \rightarrow (\lambda I-T)^{-1}$ is analytic on $\rho(T)$(which is the resolvent set). If I fix an element ...
0
votes
2answers
108 views

A question about weighted forward unilateral shift operators

We define $$ B(x_{1}, x_{2},...)=(0, \frac{x_{1}}{2}, \frac{x_{2}}{3},...,x_{n})\in l^{2}(N), $$ How could be shown that that $B$ is a quasinilpotent?
2
votes
0answers
54 views

A characterisation of the cyclic subfactors by the existence of a cyclic vector?

A cyclic subfactor is a subfactor admitting a distributive intermediate subfactors lattice. Let's start with the finite index irreducible depth 2 subfactors, i.e. the class of subfactors of the form ...
1
vote
0answers
46 views

A question about tensor product of algebras of compact operators. [duplicate]

Let $\cal{H}$ be a separable Hilbert space and $\cal{K(\cal{H})}$ the algebra of compact operators acting on $\cal{H}$. Then $$\cal{K(\cal{H})}\otimes\cal{K}(\cal{H})\cong\cal{K}(\cal{H}\otimes H).$$ ...
4
votes
1answer
100 views

Two question on a lemma about C*-algebra

I am reading Lin Hua xin's book "An introduction to the classification of amenable C*-algebras" and i am confused with the lemma 1.7.12 in this book. Lemma 1.7.12 Let $A$ be a C*-algebra and $f\in ...
2
votes
0answers
33 views

What is Kadison's process about cocycles?

My teacher told me the Kadison's process(may be not this ward, it is just my translation ) can make a 2-cocycle turn to be a cocycle(i.e.,derivation). But I can not find it in the internet. Thanks a ...
6
votes
2answers
119 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 ...
1
vote
1answer
56 views

A question about strongly continuous.

I am reading a book about C*-algebra. In the book, Let $\phi$ be a linear functional on $B(H)$ ($H$ denotes a Hilbert space), if $\phi$ is strongly continuous, therefore, there exist vectors ...
1
vote
2answers
61 views

A question about a linear bounded operator (in hilbert space)

I am reading a book about C*-algebra. When i study von Neumann algebras in this book, i meet with a problem. In the book, If $H$ is a Hilbert space, we write $H^{(n)}$ for the orthogonal sum of n ...
0
votes
1answer
55 views

Prove the left multiplication $L_A$ operator in $\mathcal{B}(\ell_2)$ is continuous?

Can someone assist me in showing that this operator is continuous as a map in the weak operator topology? I tried to do this with nets, but got stuck trying to "move" the operation inside the inner ...