# Tagged Questions

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 ...

38 views

### Generalized polar decomposition

Let $x\in B(H)$. We say $(x,v,y)$ is a polar decomposition for $x$ if, $\bullet$ $y$ is positive. $\bullet$ $v$ is a partial isometry with $x=vy$. $\bullet$ Ker$(x)$=Ker$(y)$=Ker($v$) The polar ...
17 views

### the standard even grading on $M_2(A)$ and $A\otimes \mathbb{K}$

I have a question about a passage in Blackadar's book about K-Theory. Let $A$ be a (ungraded) $C^*$-algebra. There is a grading on $M_2(A)$ with $M_2(A)^{(0)}$ the diagonal matrices and $M_2(A)^{(1)}$...
34 views

### If a positive operator $y$ has the same kernel as $cy$, what can we conclude about the kernel of $c$?

Let us consider the equation $x=cy$ in $B(H)$. Assume that: $y$ is a positive operator. $x$ and $y$ have the same null space. Ker($y$) is contained in Ker($c$). Can we conclude that Ker($y$)=...
128 views

21 views

### subset of pure states with norm condition already dense

I struggle to proof the following statement: Let $Y\subseteq P\left(B\right)$ a subset of pure states on a $C^*$-Algebra $B$ such that for every $b\in B$ there exists a $\varphi \in P \left(B\right)$ ...
30 views

### why is $f\otimes g:A\otimes_{\min}C\to B \otimes_{\min} D$ injective?

If $f:A\to B$, $g:C\to D$ are injective $\ast$-homomorphisms between $C^*$-algebras $A, B, C, D$, is the induced map on the spatial tensor product $$f\otimes g:A\otimes_{\min}C\to B \otimes_{\min} D$$...
30 views

### What is the motivation of studying $P[A]$ in operator K-theory?

I am reading the last chapter of Murphy's $C^*$-algebras and operator theory. He defines $$P[A]=\bigcup_{n=1}^\infty\{p\in M_n(A):\text {p is a projection} \}$$ and construct the Grothendieck group ...
24 views

### Strong convergence of Spectral Projection

Let $H$ be a Hilbert space and $B(H)$ be the space of all bounded linear operators on $H$. Assume that $\{A_n\in B(H)\}_n$ strongly converges to $A$. $E^{|A|}(1,\infty)$ is a spectral projection of ...
54 views

### Aproximating positive elements in inductive limit of C* algebras

Let $\{A_i,\Phi_{ij} \}_{i\in \mathcal{I}}$ a directed system of C* algebras and $A:=\varinjlim A_i$ its limit. I know that if $x\in A$ is self-adjoint, it can be approximated with another self-...
14 views

### Crossed product by locally finite group

If a countable discrete group $G$ is the direct limit of finite subgroups $F_i$, and $G$ acts on a compact Hausdorff space $X$, can the crossed product $C^*$-algebra $C(X)\rtimes_r G$ be described in ...
15 views

39 views

### The second isomorphism theorem for C*-Algebras

in my functional analysis class right now we are studying the basics of C* Algebras and I was recently asked this question about the second isomorphism theorem for C* Algebras, but first let me cite ...
57 views

### positive operator, projection on Hilbert,$Q|T|Q \ge |QTQ|?$
Let $T$ be an operator on a Hilbert space $H$. And $Q$ be a projection. Whether $$Q|T|Q \ge |QTQ|?$$ Obviously, if $T$ is positive, then $Q|T|Q = |QTQ|$. Also, there are some $T$ such that $QTQ=0$ ...