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 [c-star-algebras]

A C*-algebra is a complex Banach algebra together with an isometric antilinear involution satisfying $(ab)^*=b^*a^*$ and the C*-identity $\Vert a^*a\Vert=\Vert a\Vert^2$. Related tags: (banach-algebras), (von-neumann-algebras), (operator-algebras), (spectral-theory).

1
vote
0answers
37 views

GNS representation and states of $C^*$-Algebras

Let $\varphi$ be a state of the $C^*$-algebra $A$, $B\subset A$ a hereditary subalgebra and $K_\varphi:=\{x\in B : 0\le x \le 1, \varphi(x)=1\}$. Let $\pi_\varphi:A\rightarrow \mathcal{B}(H_\varphi)$ ...
3
votes
1answer
51 views

Continuous Functional Calculus Argument in Fell's Paper

I am reading an old paper of Fell's and I am having some problems sorting out one of his continuous functional calculus arguments. The essential problem is this. Let $A$ be a C$^{*}$-algebra and let $...
1
vote
0answers
24 views

Actions of $C^*$-dynamical systems on primitive ideals

Reading about induced Systems of $C^*$-Algebras, I found this one statement that I can't figure out. Let $G$ be a compact Group and $(A,G,\alpha)$ a $C^*$-dynamical System, such that for some closed ...
2
votes
1answer
36 views

Approximating states of the enveloping von Neumann Algebra

Let $A$ be a $C^*$-Algebra and $A''$ its enveloping von Neumann Algebra. Is the state space $S(A)$ of $A$ weak*-dense in the state space of $S(A'')$? I Know that every state on $A$ extends as a vector-...
1
vote
1answer
34 views

Faithful normal state on type I von Neumann algebra

Let $\mathcal{M}$ be a type I von Neumann algebra. Are there necessary and sufficient conditions for $\mathcal{M}$ to admit a faithful normal state? If I think of $\mathcal{M}$ as the von Neumann ...
2
votes
2answers
28 views

The subgroup of unitaries in $C^*$-algebra is not open

I could show that the subgroup of invertibles are open in a $C^*$-algebra, I know the fact that the subgroup of unitaries in a $C^*$-algebra is not open. I am trying to find an example. Looking for ...
4
votes
1answer
94 views

Introduction to von Neumann algebras

I'm learning the basics of von Neumann algebras. Every reference on the subject I can find turns to the study of projections, introduces factors and the type classification immediately after having ...
1
vote
2answers
46 views

'projections' on $C^*$-algebras. Under certain assumptions, is $f(a)=a$ for all $a\in A$?

Let $B$ a unital $C^*$-algebra and $A$ a $C^*$-subalgebra of $B$ containing the unit of $B$. Let $f:B\to B$ be a linear, unit-preserving, completely positive and idempotent map (i.e. $f^2=f$). ..[I ...
2
votes
0answers
43 views

Dixmier averaging Theorem

We know the standard result of Dixmier averaging Theorem for von Neumann algebras. Is Dixmier averaging Theorem still holds for $C^{*}$-algebras??
3
votes
2answers
79 views

A common lifting of a positive element?

Suppose $A$ is a C$^{*}$-algebra and that we have finite-dimensional irreducible representations $\varphi_{1},\ldots,\varphi_{n}$ of $A$. It follows from the question C$^{*}$-algebra acting ...
2
votes
1answer
60 views

Perturbation Theory: Derivative of a trace.

The problem that I am looking at is the following perturbation problem from the notes on Trace Inequalities and Quantum Entropy on page 12, the following result is said to follow "by the Spectral ...
1
vote
1answer
35 views

irreducible subrepresentation of a finite dimensional representation of $C^*$ algebra

Let $A$ be an infinite dimensional $C^*$ algebra.$\psi:A\rightarrow M_{k_1}(\mathbb{C})\oplus M_{k_2}(\mathbb{C})\oplus\ldots\oplus M_{k_r}(\mathbb{C})$ is a nonzero $*$ homomorphism,where $k_1,\...
0
votes
1answer
38 views

Is finite dimensional representation of $C^*$ algebra surjective?

Suppose $A$ is a $C^*$ algebra,$B$ is a finite dimensional $C^*$ algebra,$\phi:A\rightarrow B$ is a nonzero $*$ homomorphism. 1.Can we deduce that $\phi$ is surjective 2.Does there exist a nonzero $*...
1
vote
1answer
31 views

unital and completely isometric imply completely positive?

Let $B, C$ be unital $C^*$-algebras, $S\subset C$ an operator system, $f:S\to C$ a unital linear map. Then, $f$ is completely isometric if and only if $f$ is isometric and both $f$ and $f^{-1}:f(S)\...
2
votes
2answers
90 views

How to calculate the $K_0$ and $K_1$ groups for $A$

Let $A=\{f\in C([0,1],M_n)\mid f(0)$ is scalar matrix $\}$. Then find the $K_0(A)$ and $K_1(A)$. I am trying to use the SES $J \rightarrow A \rightarrow A/J$ where $J$ can be taken as some closed ...
1
vote
0answers
40 views

Separable Commutative $C*$ algebra [duplicate]

Consider the algebra $C(X)$ of continuous complex functions over a compact space $X$. On what conditions this algebra is separable? What if $X$ is a compact subset of $\mathbb{R}^n$?
4
votes
1answer
61 views

Injective homomorphism $H:C_b(X) \to C_b(Y)$ implies the existence of a continuous and surjective map $F:Y \to X$

Let $X$ and $Y$ be $2$ topological spaces and let $C_b(X)$ and $C_b(Y)$ denote the set of all continuous and bounded functions on X and Y, respectively, to the space of complex numbers. It is a well-...
2
votes
2answers
57 views

Show a relation for the state on $C^*$ algebra

Let $\varphi$ be a state on $C^*$-algebra $A$. Assume $\varphi(a^2)=\varphi(a)^2$ for some self-adjoint elements $a\in A$. Show that $\varphi(ab)=\varphi(ba)=\varphi(a)\varphi(b)$ for any element $b\...
1
vote
1answer
21 views

Two ordered relations on projections.

Let $A$ be a vn-algebra. Let $p$ and $q$ be two projections. In the literature, we say $p$ is majorised by $q$ if $pq=p$. Q. Suppose that $q-p$ is a positive element in $A$ (meaning $q-p=x^*x$ for ...
1
vote
1answer
61 views

Show that the space of continuous functions on the compact Hausdorff space with matrix-value is a $C^*$-algebra

Given $X=0 \cup \{1/n\}$. We can show that it is a compact Hausdorff space. Now take $M_n$ to be the matrix algebra. I am confusing on showing that both $C(X, M_2)$ and $B=\{f \in C(X, M_2)$ where $f(...
2
votes
1answer
39 views

Pure states on subalgebras of $\mathcal{B}(\mathcal{H})$ in finite dimensions.

I consider only finite-dimensional Hilbert spaces. We know that pure states on $\mathcal{B}(\mathcal{H})$ are exactly the vector states or in terms on density matrices, the rank one projections. My ...
1
vote
1answer
35 views

the induced trivial $*$ homomorphism

Let $A=c_{0}\oplus \mathbb{K}$,$I=c_{0}$ is the closed ideal of $A$,there is an induced $*$ homomorphism $\phi:A/I\rightarrow M(I)/I$,where $M(I)$ is the multiplier algebra of $I$.$\phi(a+I)=(L_{a},R_{...
2
votes
0answers
47 views

Union of $C^*$-algebras generated by one element also generated by one element?

Let $A$ be a unital $C^*$-algebra and $a_1, a_2, ... \in A$ be elements such that $C^*(a_1, 1) \subseteq C^*(a_2, 1)\subseteq ...$ where $C^*(., 1)$ denotes the generated unital $C^*$-algebra. The ...
1
vote
1answer
45 views

Upgrading Injectivity of a *-homomorphism From a Dense Subalgebra

In the proof of Lemma A.4 of this document, the author proves that a C$^*$-algebra is simple by showing that every one of its representations is faithful. They proceed by taking a representation and ...
0
votes
1answer
45 views

Norm inequality for linear combination of non-commuting unitaries

Let $u, v$ be unitaries in a unital $C^*$-algebra satisfying $uv=e^{2\pi i \theta}vu$ where $\theta$ is irrational (so $\{e^{2 \pi i n \theta} : n \in \mathbb{Z} \}$ is dense in $\mathbb{T}$). For ...
2
votes
1answer
79 views

Definition of a closed ideal in a $C^*$ algebra

Let $A$ be a commutative and unital $C^*$-algebra. What is the definition of a closed ideal of $A?$ My understanding: A subset $I$ of $A$ is an ideal if it is a vector subspace of $A$ and for any $...
0
votes
1answer
34 views

Does the algebra of real valued functions in Stone-Weierstrass Thm. equal to the set of polynomials?

I am studying Stone-Weierstrass Theorem. I wonder whether A is equal to the set of polynomials? If so, how can I proof this? And the statement is as follows: Let $S$ be a compact set, and let $A$ ...
2
votes
1answer
42 views

Numerical range of selfadjoint elements in non-unital C*-algebras

If $a$ is an element of a C*-algebra $A$ then $V(a)=\{\varphi(a): \varphi\text{ is a state of }A\}$ is the numerical range of $a$. If $a$ is selfadjoint and $A$ is unital then it is known that $V(a)=[\...
5
votes
3answers
83 views

What's wrong with my proof that $\sigma(a)\subseteq[-\|a\|, \|a\|]$ for $a$ self-adjoint?

Let $U$ be a $C^*$-algebra and $a\in U$ be self-adjoint. I have a simple proof that $\sigma(a)\subseteq [-\|a\|,\|a\|]$, where $\sigma(a)$ is the spectrum of $a$. It goes as follows (the facts used ...
1
vote
1answer
47 views

the uniform norm of a positive linear map on von Neumann algebras

Let $H$ be a Hilbert space. Let $\phi:B(H)\to B(H)$ be a positive linear map. Q. Do we have $\|\phi\|=\sup\{\|\phi(x)\| : 0\leq x\leq1\}$?
0
votes
1answer
41 views

Finite dimensional von Neumann algebra [closed]

How to prove that finite dimensional von Neumann algebra is direct sum of matrix algebras?
1
vote
1answer
57 views

minimal projections in finite dimensional von Neumann algebras

The algebras I'm working with are defined as follows Let $\mathcal{H}$ be a Hilbert space of finite dimension and denote by $\mathcal{B}(\mathcal{H})$ the bounded operators on $\mathcal{H}$. A ...
1
vote
1answer
33 views

Monic projections in finite von Neumann algebra

The observation and heart of the proof of existence of trace lies on the fact in finite vN algebra any projection is orthogonal sum of monic projections, can somebody reveal me the idea and motivation ...
2
votes
1answer
70 views

Is the double commutant $A''$ commutative if $A$ is commutative?

If $A$ is a commutative C*-subalgebra of linear bounded operator space $B(H)$ on some Hilbert space $H$, so is the double commutant $A''$. It follows from $A$ is dense in $A''$ and the multiplication ...
1
vote
2answers
29 views

Bounding a positive element from below using a dense subset

Let $A$ be a $C^*$-algebra. Let $a$ be a nonzero positive element of $A$. Suppose that $A$ equals the closed span of a subset $B$ of $A$, where $B$ is closed under the $*$ operation, linear ...
0
votes
1answer
18 views

Does there exist a connection between contractive completely positive map and surjective map

If $\psi:A \rightarrow M_n(\mathbb{C})$ is a c.c.p map.What is the relationship between c.c.p maps and surjective maps?Can we deduce that $\psi$ is a surjective map?If not,does there a close ...
3
votes
1answer
53 views

State space of finite dimensional, abelian C*-algebra is a simplex.

I am looking for a proof that the state space of a finite dimensional C*-algebra is a simplex and, vice versa, if the state space is a simplex, the C*-algebra is abelian. I've found one proof, but it ...
-1
votes
1answer
18 views

limit of sequence of bounded operators

Suppose $\phi_n:A\rightarrow B(H_n)$ is a sequence of nonzero representations, where $A$ is a nonunital $C^*$-algebra,$H_n$ is a Hilbert space, and $P_n$ is a sequence of projections on $H_n$. Does ...
1
vote
1answer
42 views

Identifying tensors with functions in $C^*$ -algebras

We know the result $C(Y,C(X)) \cong C(X)\otimes C(Y)$, I don't able to construct the isomorphism mapping that by starting an arbitrary function $f$ from $C(X,Y)$, how to get tensor element in $C(X\...
0
votes
0answers
33 views

Structure theory of type 1 von Neumann algebras

Why structure theory of Type 1 von Neumann algebras are coming from spectral theorem? I read Arveson, their heavy technical things are used of multiplicity theory (Hahn-Hellinger Theorem) to get the ...
1
vote
1answer
47 views

If $r(a) < 1$, does $\sum_{n=0}^\infty a^{*n}a^n$ necessarily converge?

In Murphy, exercise 2.6: Let $A$ be a unital C$^*$-algebra. If $r(a) < 1$ and $b = (\sum_{n=0}^\infty a^{*n}a^n)^{1/2}$, show that $b \geq 1$ and $\lVert bab^{-1}\rVert < 1$.... where $r(a)$ ...
0
votes
1answer
74 views

Convergence of a net in a Hilbert space

Suppose $\phi:A\rightarrow B(H)$ is a nonzero $*$ homomorphism, where $A$ is a nonunital $C^*$ algebra, $H$ is a Hilbert space, $\{x_{i}\}$ is a net of unit vectors in $H$, does there exist $a_0\in A$ ...
2
votes
1answer
57 views

Representations of simple C$^*$-algebras

I am reading from the following document, and am a bit stumped by footnote 4 on page 5: https://arxiv.org/pdf/math-ph/0006011.pdf Actually, I will copy the relevant text because it disappears off ...
2
votes
1answer
104 views

Why do we need the spectral theorem? What is its purpose?

One realization of spectral theorem for me that we want to make sense "the object $:f(T)$" in von Neumann algebra $M$ where $f$ is bounded measurable function with respect to some measure. ...
2
votes
0answers
36 views

The Image of Unitary Representation in the Space of Bimodules

TL;DR: Bimodules over a von Neumann algebra are commonly understood as a generalization of group representation. Indeed, when the von Neumann algebra $N$ is $\mathcal{L} G$, the unitary ...
1
vote
1answer
41 views

$*$ homomorphism $\phi$ from $A$ to multiplier algebra $M(I)$

If $I$ is a closed ideal in $C^*$ algebra $A$, then there is a unique $*$ homomorphism $\phi$ from $A$ to $M(I)$ which extends the $*$ homomorphism $I\to M(I)$, where $M(I)$ is the multiplier algebra ...
3
votes
1answer
41 views

Does the set of characters, $\Omega(\mathcal{A})$, over a C${}^{\ast}$-algebra, $\mathcal{A}$, generate a weakly dense subspace of $\mathcal{A}'$?

Let $\mathcal{A}$ be an abelian C${}^{\ast}$-Algebra with unit. We know that $\mathcal{A}\cong C(\Omega(\mathcal{A}))$, where $\Omega(\mathcal{A})\subseteq\mathcal{A}'_{\geq 0}$. Note that for $\...
1
vote
1answer
28 views

infinite representation of a $C^*$ algebra

Suppose $\pi$ is a finite dimensional representation of $C^*$ algebra $A$ on a Hilbert space $H$,then $\pi$ is the direct sum of finite dimensional irreducible subrepresentations. My quesion is :If $\...
1
vote
1answer
52 views

Some particular example clarification on algorithm of Hahn Hellinger Theorem

Consider the self adjoint matrix $T$ \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 &0 &2 \end{bmatrix} The question is the following: I want to understand the ...
-2
votes
1answer
42 views

Can $C^*$ algebra $A$ be decomposed?

If $A$ is a $C^*$ algebra ,$B$ is a finite dimensional $C^*$ subalgebra of $A$.Does there exists a $*$ subalgebra of $C$ such that $A=B \oplus C$?