# 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).

1,436 questions
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### 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 $\pi_\varphi:A\rightarrow \mathcal{B}(H_\varphi)$ ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### $*$ 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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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 $\... 1answer 28 views ### infinite representation of a$C^*$algebra Suppose$\pi$is a finite dimensional representation of$C^*$algebra$A$on a Hilbert space$H$,then$\pi$is the direct sum of finite dimensional irreducible subrepresentations. My quesion is :If$\...
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 ...
### 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 $A=B \oplus C$?