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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).

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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 $*...
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1answer
10 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)\...
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2answers
54 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 ...
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0answers
29 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 $\mathbb{R}^n$?
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1answer
50 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-...
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2answers
54 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\...
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1answer
16 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 ...
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1answer
46 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(...
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1answer
36 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 ...
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1answer
29 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_{...
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0answers
36 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 ...
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1answer
35 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 ...
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0answers
38 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 ...
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1answer
40 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 $...
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1answer
33 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 $A$ ...
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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)=[\...
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3answers
76 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 ...
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1answer
46 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 $\|\phi\|=\sup\{\|\phi(x)\| : 0\leq x\leq1\}$?
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1answer
36 views

Finite dimensional von Neumann algebra [closed]

How to prove that finite dimensional von Neumann algebra is direct sum of matrix algebras?
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1answer
45 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 ...
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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 ...
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1answer
58 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 ...
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2answers
28 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 ...
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1answer
15 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 ...
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1answer
45 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 ...
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1answer
16 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 ...
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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\...
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0answers
31 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 ...
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1answer
46 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 $r(a)$ ...
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1answer
69 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 $a_0\in A$ ...
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1answer
52 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 ...
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1answer
99 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. ...
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0answers
34 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 ...
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1answer
32 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 ...
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1answer
38 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 $\...
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1answer
24 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 $\...
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1answer
50 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 ...
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1answer
40 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 $A=B \oplus C$?
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1answer
46 views

splitting lemma in $C^*$ algebras

In abelian category,there is a splitting lemma. see https://en.wikipedia.org/wiki/Splitting_lemma I wonder whether the splitting lemma also holds in $C^*$ algebras .Is left split equivalent to right ...
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1answer
45 views

Is functional calculus continuous on elements of the algebra.

Suppose $A$ is a C*-algebra, $a$ is a hermitian element of $A$. For each continous function $f:\mathbb{R}_+\to \mathbb{C}$, we say $f$ is continuous on $A$ if for every sequence $\{a_\lambda\}$ of ...
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1answer
35 views

tracial extension on a $C^*$ algebra

If $A$ is a $C^*$ algebra, I konow the fact :If $\tau$ is a state ,we can extend it to obtain a state on$M(A)$ by continuity of $\tau$,where $M(A)$ is the multiplier of $A$. If $\tau$ is a tracial ...
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1answer
26 views

description of an ideal generated by the projections in a $C^*$ algebra

If $A$ is a $C^*$ algebra,$P_i$ are projections in $A$,$I$ is the ideal generated by the projections.I think $I$ is the $C^*$ algebra generated by $P_iAP_i$.How to charectarize $I$,is there a precise ...
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1answer
20 views

tracial states on a finite dimensional $C^*$algebra

If $A$ is a finite dimensional $C^*$ algebra,how many tracial states on $A$ ,is it countable or uncountable?How to construct a tracial state on $A$? Can anyone give me some hints?Thanks.
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1answer
29 views

the $C^*$ algebra generated by $P_nAP_n$ is isomorphic to $c_0$ direct sum of the algebras $P_nAP_n$

Suppose $A$ is a $C^*$ algebra,if $P_n$ are pairwise orthogonal central projections in $A$. Then the $C^*$ algebra generated by $P_nAP_n$ is isomorphic to $c_0$ direct sum of the algebras $P_nAP_n$. ...
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2answers
40 views

finite dimensional representation of $C^*$ representation

According to the standard definition of representation of a $C^*$ algebra $A$ ,we need to construct a $*$ homomorphism $\pi$ from $A$ to $B(H)$,where $H$ is a Hilbert space.My question is : if we have ...
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1answer
40 views

Reference request: extension of $*$-homomorphism to multiplier algebra

Let $A$ be a $C^*$-algebra and $f:A\rightarrow\mathbb{C}$ a $^*$-homomorphism. Does $f$ always extend to a $^*$-homomorphism $\tilde{f}:M(A)\rightarrow\mathbb{C}$, where $M(A)$ is the multiplier ...
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1answer
18 views

Existence of nonfaithful tracial states in infinite von Neumann algebra

Can there exist non faithful tracial state in infinite von Neumann algebras ? As faithfulness plays the role to show the vN algebra has to be finite. Further are all hyperfinite $II_{1}$ factors are ...
2
votes
1answer
29 views

If $A$ is a unital direct limit of C*- algebras, why can we assume that the connecting maps are unital? [closed]

I know that we may assume that each $\phi _{n}$ is injective. Then how to show that we may assume $\phi_n$ is unit preserving when $n \geq N$ for some $N$ ?
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1answer
60 views

GNS representation doubt

Let $M$ be a von Neumann algebra inside $B(\mathcal{H})$, if we take $\xi$ $\in \mathcal{H}$, consider vector state $\omega_{\xi}$, $(i)$ Does there exist GNS Hilbert space coming from $M$ such that ...
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votes
1answer
41 views

Faithful normal states on C-Star algebras [closed]

Let $A$ be a c-star algebra acting on a non separable Hilbert space. Can one always define a faithful normal state on it?