Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [operator-algebras]

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 ...

-1
votes
0answers
21 views

commutant of a non unital $C^*$ algebra

If $A$ is a non-unital $C^*$ algebra,is the commutant of $A$ empty? Does there exist a theorem which states that every $C^*$ algebra has commutant(centralizer)?
1
vote
1answer
14 views

Non normal state

Does anybody know an example of a state on a von Neumann algebra that is not normal? If it has relevance to physics it would be nice.
0
votes
0answers
8 views

The dual of $C_0(X,L^1(Y,\alpha))$

I'm reading "Anantharaman-Delaroche, C.; Renault, J. Amenable groupoids." (https://mathscinet.ams.org/mathscinet-getitem?mr=1799683), and more specifically the proof Proposition 1.1.5. I will write ...
0
votes
2answers
22 views

Does norm equivalence imply norm equivalence of induced operator norms?

Let $X$ be a complex vector space and let $\|\cdot\|_1$ and $\|\cdot\|_2$ denote norms on $X$ which are equivalent, i.e. there exist constants $c,C > 0$ such that for all $x \in X$ $$ c \|x\|_1 \...
0
votes
0answers
18 views

multiplication of bounded linear operators

Suppose $T_1,T_2\in B(\oplus H_n)$,each $H_n$ is an infinite dimensional Hilbert space,I want to construct $S_1,S_2\in B(\oplus H_n)$ such that $S_1T_1=T_2S_2 =(0\cdots,k I_n,\cdots,0)$,where k is ...
0
votes
1answer
22 views

multiplication of Hilbert Schmidt operators and bounded operators

Suppose $T_1,T_2 \in B(H)$,where $H$ is an infinite dimensional Hilbert space. $S\in \mathcal{HS}(H)$,$\mathcal{HS}(H)$ is the set of Hilbert Schmidt operators on $H$. Does there exist nonzero ...
0
votes
0answers
35 views

inverse of a bounded linear operator [on hold]

Suppose $T\in B(H)$,where $H$ is an infinite dimensional Hilbert space. If $\|I–T\|<1$,then $T$ is invertible. 1.Does there exist other methods to determine whether $T$ is invertible or not. 2....
2
votes
0answers
14 views

Predual of $W^*$-subalgebra

I've seen many references claiming that if $\mathcal{N}$ is a $\sigma$-weakly closed *-subalgebra of a von Neumann algebra $\mathcal{M}$, then by taking $\mathcal{N}_\bot:=\{\phi\in\mathcal{M}_*|a(\...
2
votes
0answers
22 views

Schwarz inequality for unital positive maps on C*-algebras

I was recently studying this paper by Man-Duen Choi about inequalities for positive maps on C*-algebras. He demonstrates that Let $\phi : \mathcal{A} \to \mathcal{B}$ be a unital positive linear ...
1
vote
0answers
38 views

Proof that regular representation induced by faithful representation is faithful on reduced crossed product

Suppose that $A$ is a $C^*$-algebra and $G$ is a locally compact Hausdorff group acting on $A$. Gert Pedersen, in his book $C^*$-algebras and their automorphism groups, shows in theorem 7.7.5 that ...
1
vote
1answer
31 views

Operator norm of a family of matrices

Let $c$ be a complex number. Consider the family of $n\times n$ matrices $M_n$ which have $c$'s on one off-diagonal, $\bar{c}$'s on the other off-diagonal, and zero everywhere else. So $M_4$ looks ...
1
vote
2answers
35 views

state on a non-unital $C^*$ algebra

Suppose $\tau$ is a state on a non-unital $C^*$ algebra $A$.There is a well-known inequality: $$\tag{$*$}|\tau(a)|^2\leq\tau(a^*a),\ \text{ for all } a\in A.$$ Does there exist some nonzero element $...
1
vote
2answers
26 views

Can a wo-converging sequence of bounded operators on a Hilbert space be NOT uniformaly bounded?

Let $\{T_n\}$ be a sequence of bounded operators on a Hilbert space. Assume that $T_n\rightarrow T$ in weak operator topology. Is this sequence necessarily uniformly bounded? It seems that if this ...
0
votes
0answers
26 views

faithful tracial states on a $C^*$ algebra [closed]

Does there exist a theorem or a proposition to determine whether a $C^*$ algebra has faithful tracial state or not?
-1
votes
3answers
53 views

Complementary $C^*$-subalgebra

Does there exist a closed subspace of $\ell^{\infty}$ complementary to $c_0$? If $A$ is a $C^*$-algebra, $B$ is a $C^*$-subalgebra of $A$, under which condition can one ensure that there exists a $C^*...
2
votes
1answer
33 views

Nuclear $C^*$ algebra and tensor products

Suppose $A,B$ are $C^*$ algebras, $\alpha$ is some $C^*$-norm on the algebraic tensor product of $A$ and $B$. If $A\otimes_{\alpha}B$ is nuclear, can we conclude that $A$ and $B$ are nuclear? What ...
2
votes
1answer
37 views

unique $*$ homomorphism of spatial tensor product

If $A$ is a nuclear $C^*$ algebra, $A^{op}$ is the opposite $C^*$ algebra, is the conclusion: "there is a unique injective $*$-homomphism $\pi \colon A\otimes A^{op}\rightarrow M(A)\otimes M(A)^{op}$" ...
0
votes
2answers
54 views

Elementary tensors of tensor product of C*algebras

When $\alpha$ is a $C^*$-norm on $A \times B$, we denote the $C^*$ completion of $A \otimes B$ with respect to $\alpha$ by $A\otimes_{\alpha}B$. I feel a little confused about the elementary tensors ...
0
votes
1answer
35 views

construct a faithful tracial state throgh a tracial state on some $C^*$ algebra [closed]

Suppose $A$ is a $C^*$ algebra ,$\phi$ is a tracial state (not faithful)on $A$.Can we construct a faithful tracial state on$A$ by using $\phi$?
3
votes
1answer
51 views

$C^*$-algebra norm computations

I'm fairly new to $C^*$-algebras and Hilbert space. Given the algebraic relations of the $C^*$-algebra, I am having a lot of trouble computing the norm of its elements and am wondering if there are ...
2
votes
1answer
46 views

product states on the tensor product *-algebra

Let A and B be two unital $C^∗$-algebras, and $x∈A⊗B$ (the algebraic tensor product $*$-algebra), different from $0$. Is there states $ω_x∈A^∗_+$ and $φ_x∈B^∗_+$ such that, for the product state $ω_x×...
2
votes
1answer
34 views

Is there an example of a non-zero projection in a C$^{*}$-algebra that is infinite but not properly infinite?

For clarification: Given a projection $p$ in a C$^{*}$-algebra $A$, we say $p$ is infinite if there is a projection $q\in A$ satisfying $q\lneq p \sim q$; we say $p$ is properly infinite if there are ...
0
votes
0answers
17 views

tensor product of multiplier algebras

Let $A$ be a $C^*$ algebras,$A^{op}$ is the opposite $C^*$ algebra. Can we view $A \otimes_{\alpha} A^{op}$ as a $C^*$-subalgebra of $M(A)\otimes_{\alpha} M(A)^{op}$ ,where $\alpha$ is any $C^*$ norm,$...
4
votes
1answer
40 views

$GL_n^+(A)$ is open but $U_n^+(A)$ is not

Let $A$ be a C*-algebra. $\tilde{A}$ be the unitization of $A$. I checked the following lemma: If $x$ and $y$ are elements of $M_n(\tilde{A})$ such that $x$ is invertible and $\|x-y\| \leq \frac{1}{...
2
votes
1answer
49 views

Is the extension of $*$ homomorphism unique?

If $\phi:A\to B(H)$ is a $*$ homomorphism, do there exist two different $*$ homomorphisms $\phi_1,\phi_2:M(A)\to B(H)$ which extend $\phi$, where $M(A)$ is the multiplier algebra of $A$?
2
votes
1answer
41 views

Irreducible *-representations of $M_n(\mathbb{C})$

This might be a very stupid question but, is the defining representation the only irreducible *-representation of $M_n(\mathbb{C})$? This seems to me that this should be the case but I would have no ...
0
votes
0answers
27 views

On existence of Singular value decomposition in von Neumann algebras

For which class of von Neumann algebras we will have singular value decomposition of each element of the algebra?
0
votes
0answers
28 views

Regarding direct limit of C* algebra

I am trying to understand direct limit in category of $C^*$ algebras. Is it well known that direct limit behaves well with double dual and quotient of $C^*$ algebras? Any references or ideas?
2
votes
0answers
72 views

Sufficient condition for element to be close to an invertible in Rordam's Simple C$^{*}$-algebras paper

I am reading Rordam's paper "On the Structure of Simple C$^{*}$-Algebras Tensored with a UHF-Algebra." I came across the following passage: I am having trouble understanding the $(\impliedby)$ ...
2
votes
1answer
22 views

Isomorphism between $\mathcal{L}_{M_n(A)}(E^n)$ and $\mathcal{L}_{A}(E^n)$

The following doubt came after reading the book "Hilbert C*-modules" by E.C. Lance. Let $A$ be a C*-algebra and $E$ a Hilbert $A$-module, there's a natural structure of Hilbert $A$-module on $E^n$ ...
0
votes
0answers
32 views

First countable dual space of a von Neumann Algebra.

Let A be a C* algebra on a Hilbert space H and R=A'', the respective von Neumann algebra. If A is separable, the set of its states is a first countable topological space in the weak* topology. Does ...
1
vote
1answer
30 views

Bases of the tracial cone and full elements

Say $A$ is an exact C*-algebra and let $T(A)$ be the cone of densely defined lower semicontinous traces. It is known that if $a \in \mathrm{Ped}(A)$ is full, then $T_{a\to 1} := \{\tau \in T(A) \mid \...
0
votes
1answer
32 views

weak^* uniform convergence compact

I have a question about a definition from the book of Gert Pedersen, $C^*$-algebras and their automorphism groups. In the book setup, we have a locally compact group $G$ and a $C^*$-algebra $A$ and ...
4
votes
1answer
47 views

Dependence of operator topologies in a $C^*$ algebra on the representation

Let $A$ be a $C^*$ algebra. Given a faithful representation $\pi:A\to \mathcal{B}(H)$, we can define the weak operator topology with respect to $\pi$ as initial with respect to the maps $a\mapsto \...
1
vote
1answer
46 views

why this sequence is exact?

let $A$ denotes the Toeplitz algebra, $u$ is the unilateral shift, let $\tau$ be the unique *-homomorphism such that $\tau (u)=1$, let $A_{0}$ be the kernel of $\tau$. And $S$ is the closed ideal of $...
2
votes
1answer
36 views

Sufficient conditions for a C* algebra to be separable

Do you know of any (necessary and) sufficient conditions for a C* algebra to be separable? Reference to bibliography is welcome.
1
vote
1answer
23 views

Conditional expectation that preserve involution

Consider $B$ a C*-algebra and $A$ a C*-subalgebra such that $1_A=1_B$. If $E:B\rightarrow A$ is a faithful conditional expectation (that is a projection of norm 1, by Tomiyama's theorem) then is it ...
2
votes
1answer
27 views

the commutant of $\mathcal{HS(H)}$

Let $\mathcal{HS}(H)$ be the set of Hilber Schmidt operators on a Hilbert space,it is a $C^*$ algebra.I wonder whether we have an explicit description of the commutant of $\mathcal{HS(H)}$.Is the ...
0
votes
1answer
27 views

construct a closed subspace of $\mathcal{HS}(H)$ such that all elements in the subspace are commutative.

Suppose $K=\mathcal{HS}(H)$,where$\mathcal{HS}(H)$ is the set of all Hilbert Schmidt operators on the Hilbert space $H$.I have two questions. 1.Can we construct a closed subspace $K_1$ of $K$ such ...
1
vote
0answers
28 views

Tensor product of dual spaces and dual space of tensor product

Let $\mathcal{A}$ and $\mathcal{B}$ be infinite-dimensional C*-algebras, and let $\mathcal{A}^*$ and $\mathcal{B}^*$ denote the space of norm-continuous linear functionals on $\mathcal{A}$ and $\...
0
votes
1answer
37 views

multiplication of Hilbert Schmidt operators

Suppose $H$ is an infinite dimensional Hilbert space,$B(H)$ is the set of bounded operators on $H$,$\mathcal{HS}(H)$ is the set of Hilbert-Schmidt operators on $H$. I have two questions: 1.If $T$ is ...
0
votes
0answers
38 views

A Hilbert space is separable if and only if it admits a countable orthonormal basis [duplicate]

According to Wikipedia: A Hilbert space is separable if and only if it admits a countable orthonormal basis While this statement seems very reasonable, it is not clear to me how one would go about ...
0
votes
0answers
31 views

Reflection operator as a product of inversion and rotation

I found an statement that says the following (Merzchbacher, Quantum Mechanics): We can also proof that the reflection of a system with respect to a plane with normal vector $\vec{n}$, can be ...
1
vote
0answers
20 views

bimodule over a non-unital $C^*$ algebra

Let $A$ be a $C*$ algebra,suppose $\mathcal{H}$ is a $M(A)–M(A)$ bimodule,where $M(A)$ is a multiplier algebra of $A$.Can we deduce that $\mathcal{H}$ is a $A-A$ bimodule? My thought:since $\mathcal{...
2
votes
0answers
27 views

Hilbert subspaces whose vectors are each cyclic for a von Neumann factor

Let $\mathcal{R}$ be a type III factor acting on separable Hilbert space $H$, and $S \subseteq H$ a closed linear subspace such that every nonzero $v \in S$ is cyclic for $R$; can there exist a ...
1
vote
1answer
26 views

tracial states on $C^*$ subalgebra

Let $A$ be a $C^*$ algebra,$A=A_1\oplus A_2$.If $A$ has tracial state $\tau$,I want to show $A_1$ also has tracial state,say $\tau_1$. My thought: let $\tau_1(a_1)=\tau(a_1,0)$,where $a_1 \in A_1$,...
2
votes
0answers
59 views

Proving a Certain Inequality Without Utilizing the Full Theory of $ C^{\ast} $-Tensor Products

Suppose that we have the following objects: $ X $ — a locally compact Hausdorff space. $ A $ — a $ C^{\ast} $-algebra. $ \pi $ and $ \rho $ — commuting representations of, respectively, $ {C_{0}}(X) $...
2
votes
1answer
16 views

Show that a specific linear map from a Hilbert module into a C*-algebra satisfies a certain equation

Let $A$ be a C*-algebra, $\mathcal{H}$ be a Hilbert $A$-module and $B$ be a C*-algebra. Let $\pi: A \to B$ be a $*$-homomorphism and $\tau: \mathcal{H}\to B$ be a linear map such that $\tau(\xi)^*\tau(...
1
vote
1answer
61 views

tracial states on corona algebra

Let $A$ be the $c_0$ direct sum of $M_{n}(\mathbb{C})$,I know the fact that the multiplier algebra of $A$ ,M($A$) is $\prod M_n(\mathbb{C})$. Does the corona algebra $M(A)/A$ have uncountable tracial ...
0
votes
1answer
21 views

existence of finite irreducible reprentation of a nonunital $C^*$ algebra

Suppose $A$ is a non-unital $C^*$ algebra,can we conclude that there must exist a nonzero finite irreducible representation of $A$.