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).
2,834
questions
1
vote
1
answer
22
views
Given a representation, trying to find a non degenerate representation without changing kernel
Let $A$ be a $C^{\ast}$-algebra and $\sigma : A\to\mathcal B(H)$ be a degenerate representation of $A$ such that $H_0 := \overline{\sigma(A)H}\neq H$, then letting $H_1 := H_0^\perp$ (so that $H = H_0\...
- 3,664
2
votes
1
answer
53
views
Are there free ternary ring of operators?
I am interested in separable ternary rings of operators. For separable $C^*$-algebras we have the maximal group C*-algebra of the free group on countably many generators that quotients onto every ...
- 16.5k
2
votes
1
answer
44
views
Motivation for primitive ideals of a C*-algebra
I have recently learned about the primitive ideals and prime spectrum of a C*-algebra. I am looking for a 'reason' for why they are useful. I mean this in the sense that if I was a mathematician ...
- 591
1
vote
1
answer
38
views
Approximate identity and positive element condition
Let $u_\lambda$ be an approximate identity of a $C^*$-algebra $A$.
If $A$ has an identity $I$, a nonzero selfadjoint element $a$ is positive if and only if
\begin{equation}
\left\|I - \frac{a}{\|a\|}\...
- 23
1
vote
1
answer
47
views
Unique extension of $*$-representation into an abstract multiplier algebra
I'm trying to find a proof of the following fact:
Let $A,B$ be $C^{*}$-algebras and $\pi: A \longrightarrow M(B)$ be a non-degenerate homomorphism in the sense that $\pi(A)B$ densely spans $B$. Then ...
- 1,399
1
vote
1
answer
22
views
Projection in a hereditary subalgebra of a purely infinite C*-algebra
Let $A$ be a simple non-zero purely infinite C*-algebra. Let $p\in A$ be a projection. Then $E=pAp$ is a hereditary subalgebra. Since $A$ is purely infinite, $E$ has an infinite projection $q$.
Q. Is $...
- 337
2
votes
1
answer
61
views
Finite dimensional Irreps (of algebras) with same traces must be equivalent ('page 136' in Bourbaki)
I look for the reference (or proof) of the following fact which is from appendix (B $27$) of Dixmier's book on $C^*$-algebras.
Claim: Let $A$ be an algebra (not necessarily commutative) over a field $...
- 1,589
2
votes
1
answer
51
views
Are the Hermitian linear functionals of a $C^{\ast}$-algebra necessarily bounded?
Let $\mathcal{A}$ be a (complex) $C^{\ast}$-algebra, and let $\varphi$ be a linear functional of $\mathcal{A}$. $\varphi$ is said to be hermitian if and only if $$\forall x \in \mathcal{A}\text{,}\...
- 152
-1
votes
1
answer
55
views
Suppose $\phi(a) \geq 0$ for every state $\phi$ of $\mathcal{A}$. Can we conclude that $a\geq 0$.
Let $\mathcal{A}$ be a $C^{\ast}$-algebra and $ a \in \mathcal{A}$ be a nonzero self adjoint element.
Suppose $\phi(a) \geq 0$ for every state $\phi$ of $\mathcal{A}$. Can we conclude that $a\geq 0$.
- 3,664
1
vote
0
answers
61
views
$A \otimes J+ I \otimes B$ is prime ideal of $A \otimes B$.
Let $A$ and $B$ be $C^{\ast}-$ algebras and $A \otimes B$ denotes minimal(spatial) tensor product. Let $I$ and $J$ be prime ideals of $A$ and $B$ respectively. The following fact should be easy but I ...
- 3,664
0
votes
1
answer
48
views
Does the closure of product of two ideals satisfy $\overline{I_1I_2}=\overline{I_1}\ \overline{I_2}$.
Let $A$ be a $C^{\ast}$ algebra and $I_1$ and $I_2$ be two ideals in $A$. Is it true that $\overline{I_1I_2}=\overline{I_1}\ \overline{I_2}$?
It is clear that $\overline{I_1I_2} \supseteq \overline{...
- 3,664
1
vote
0
answers
59
views
Approaches to Atiayh Singer index theorem
Soon i will have to choose a scientific adviser.
Mathematicians in my university almost explicitly work on theory of ( partial ) differential equations, which i do not really like.
But there is one ...
- 11
1
vote
1
answer
42
views
$B(\mathcal{H}) $ can occur as a quotient by a primal ideal.
I am trying to understand the proof of the following statement:
Let $\mathcal{H}$ be a Hilbert space and $B(\mathcal{H}) $ be the space of bounded operators. Then there exist a $C^{\ast}$-algerba $\...
- 3,664
1
vote
1
answer
55
views
Tensor product of operator algebras $\mathcal B(H) \otimes \mathcal B(K)$
Given Hilbert spaces $H$ and $K$, denote $H \otimes K$ the Hilbert space tensor product with inner product $\langle x_1\otimes y_1, x_2 \otimes y_2\rangle = \langle x_1, x_2\rangle_H \langle y_1 , y_2 ...
- 13
1
vote
1
answer
42
views
Borel Function, von Neumann algebra, and pointwise monotone convergence
I am currently reading [$K$-Theory and $C^*$-Algebras: A Friendly Approach] by N. E. Wegge-Olsen. In p. 18 (section 1.3), there is a following sentence:
Allowing for bounded borel functions on $\...
- 358