Stack Exchange Network

Stack Exchange network consists of 175 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 [operator-algebras]

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 of the tags (banach-algebras), (c-star-algebras), (von-neumann-algebras), and (operator-theory).

1
vote
1answer
17 views

what's spectral axiom

I encounter a proposition in an article: For any $0\leq x\leq 1$ in a C*-algebra enjoying the spectral axiom, there are projections $(e_n)$ such that $$x=\sum_{n=1}^{\infty}\frac{1}{2^n}e_n.$$ ...
0
votes
1answer
14 views

center of a $C^*$ algebra

Suppose the center of $C^*$ algebra $A$ is 0,does this $C^*$ algebra constructed by simple non-unital $C^*$ algebras.To be more precise,$A$ can be only simple or direct sum of simple non-unital $C^*$ ...
0
votes
1answer
20 views

A basic question on the induced SOT topology on a subspace

I am trying to understand the concepts of operator topologies and I am struggling with the technical definitions. I asked myself a basic question: Suppose H is a Hilbert Space and consider the space ...
1
vote
1answer
10 views

a question on GNS space

If a $C^*$ algebra have a faithful tracial state ,we can construct a representation $(\pi_{\tau},H_{\tau})$ of $A$.$H_{\tau}$ can be obtained as following: Let $N=\{a\in A,\tau(a^*a)=0\}$,then $H_{\...
0
votes
0answers
15 views

Stability condition on homotopy functor

Let $F$ be a covariant, half exact, homotopy functor form the category of $C^*$ algebras to abelian groups. It satisfies the following. For every separable $C^*$ algebra $B$, the homomoprhism $e_*...
0
votes
1answer
13 views

construct a representation of a $C^*$ algebra

If $A$ has a tracial state,we can construct a representation $(\pi,H)$ by the GNS theorem. My question is: If $A$ has no tracial states,how can we construct a representation of $A$?Do there exist ...
0
votes
0answers
19 views

On cyclic and separating vector

Can we always have atleast two orthogonal joint cyclic and separating vectors when vN algebra $M$ is in standard form acting on GNS?
0
votes
0answers
20 views

Diagonal representation of positive power of a compact positive operator.

Let $T$ be a positive, compact operator on a Hilbert space $\mathscr H$. Now $T$ is diagonalizable i.e. there is an orthonormal basis $\{e_i:i\in \mathscr I\}$ and a bounded set of complex numbers (...
0
votes
0answers
30 views

A doubt on masa

Let $g$ in $L^{2}[0,1]$, let $f_{n}$ are in $L^{\infty}[0,1]$ such that $f_{n}{\rightarrow}^{\|\cdot\|_{2}} g$, from $f_{n}$ can we construct $g_{n}$ such that $g_{n}\rightarrow g$ in $\|\|_{2}$ norm ...
1
vote
0answers
24 views

Faithful Representation of von Neumann Algebras

I have questions regarding part of E. Christopher Lance. Ergodic Theorems for Convex Sets and Operator Algebras, Page 205. I am looking for a good reference explaining how the faithful representation ...
0
votes
1answer
26 views

Von Neumann algebra generated by a set

Suppose $A$ is a unital $C^*$ algebra,$\pi:A\to B(H)$ is a representation of $A$.Then the von Neumann algebra generated by $\pi(A)$ is equal to $\pi(A)^{"}$. Is the weak$*$ closure of $\pi(A)$ equal ...
0
votes
1answer
14 views

invariant subspace of a representation

Suppose $\pi:A \to B(H)$ is a representation of $A$ such that $\pi(A)K_1\subset K_,\pi(A)K_2\subset K_2$,where $H=K_1\oplus K_2$,can we conclude that there exist a projection $p\in \pi(A)^{'}$ such ...
1
vote
1answer
40 views

How is this operator well defined? $\frac{D}{(1+D^2)^{1/2} }$.

Let $D_+ = \partial_x +x, D_-=-\partial_x+x$. $$D= \begin{pmatrix} 0 & D_- \\ D_+ & 0 \end{pmatrix} $$ which acts on a dense subspace $C_c(\Bbb R) \oplus C_c(\Bbb R)$ of $L^2(\Bbb R) \...
1
vote
1answer
31 views

On states and dimensions in Von Neumann algebras

In an exercise, it is given that all the states satisfy normality. Can we prove the algebra to be finite dimensional? Maybe the premise means that it must be separable, and then I can show that it is ...
2
votes
1answer
23 views

Decomposition of self adjoint elements by positive elements

Let $a \in A$ be a self adjoint element of a $C^*$ algebra. There exists positive elements $a_+, a_-$, such that $$a=a_+ - a_{-} $$ $$a_+a_-=a_-a_+=0$$ Is the statement true? This is ...
0
votes
1answer
35 views

$\alpha^*$ and $\alpha_*$ in $C^*$ algebra and von Neumann algebra

I have questions regarding part of E. Christopher Lance. Ergodic Theorems for Convex Sets and Operator Algebras, Page 202. More precisely, in the paragraph starting with "We assume familiarity with.....
3
votes
2answers
47 views

Showing isomorphism of two $C^*$ algebras

It seems that quite a standard trick of showing two $C^*$ algebras are as follows: Let $A$ be a $C^*$ algebra $B$ another $C^*$ algebra. $A' \subseteq A$ be a subalgebra that is closed under $*$. (...
1
vote
1answer
27 views

Reduced group $C^*$ algebra inequality

This is part of the definition of constructing the reduced $C$-norm. Let $G$ be a locally compact hausdorff group, $\nu$ a Haar measure that is both left and right invariant, $\xi\in B(L^2(G))$, ...
1
vote
1answer
17 views

Maximal abelian subalgebra in generated von Neumann algebra

Let $D \subseteq A$ be an abelian C*-subalgebra of the C*-algebra $A$ where $A \subseteq B(H)$ for some separable Hilbert space $H$. Assume that the von Neumann Algebra generated by $D$ is a maximal ...
2
votes
1answer
27 views

How to apply continuous functional calculus

This is the statement I am using. Theorem 2.17, pg 34: Suppose that $A$ is a unital $C^*$ algebra and that $a$ is a normal element in $A$, then there is a $*$-isometric isomorphism $$C(\sigma(a)...
5
votes
0answers
54 views

Proof for a classifying space for $K$ Theory.

The goal of my question is to understand a bijection between $K_0(A)$ to $[C_0(\Bbb R), M_2(M_\infty(A))]_*$ $$[C_0(\Bbb R), M_2(M_\infty(A))]_*$$ is the homotopy class of graded $*$-homomorphisms. $...
1
vote
1answer
25 views

What Cayley transformation does to a $*$-homomorphism

We let $A$ be a $C^*$ algebra. We consider a grading on $A=C_0(\Bbb R) $ by even and odd functions whilst a grading on $M:=M_2(M_\infty(A))$ by diagonal and off diagonal elements given by grading ...
1
vote
1answer
23 views

Computing $K$-theory elements in a $C^*$ algebra $A$

Let $A$ be a unital $C^*$ algebra. Let $p,q$ be projections in $M_n(A)$. Then $[p]-[q]$ defines an element in $K_0(A)$. Now consider the matrices, the projections, $$ \left[ \begin{pmatrix} 1-p &...
2
votes
1answer
51 views

$K$ theory of $C^*$ algebra is different to algebraic $K$ Theory?

Is the $K_0$ group for a $C^*$ algebras $A$ same as that for the $K_0$ group of ring $A$ from algebraic $K$ theory? We assume $A$ is unital (I am not sure if this matters), i.e. what is an example ...
1
vote
1answer
51 views

How does one prove $C([0,1)\otimes A \cong C([0,1],A)$?

$C([0,1])$ is a $C^*$ algebra of complex functionals. $A$ is a $C^*$ algebra. Hence $C([0,1],A)$, the continuous functions from $[0,1]$ to $A$ is also a $C^*$ algebra. We construct its tensor ...
1
vote
1answer
26 views

Homotopy between unitary element and identity elements, Operator Theory

Let $\mathcal{T}$ be the Toeplitz algebra. I.e. the $C^*$ algebra generated by the shift operator $S\in B(l^2(\Bbb N))$. In page 6, line 8 of a proof we have a unitary element $u \in \mathcal{T} \...
5
votes
1answer
56 views

A short exact sequence of $C^*$ algebras

Ex 4.10.15 Let $J$ be an ideal in $C^*$ algebra $A$. Let $C:=C(A,A/J)= \{(a,f) : f:[0,1] \rightarrow A/J, f(0)=0, f(1)=\pi(a)\}$ be mapping cone of quotient map. Let $Z:=C(C,A) = \{ ((a,f),...
0
votes
0answers
16 views

A $*$-homotopy between $C^*$ algebras.

Let $J$ be an ideal of $C^*$ algebra $A$. Consider the $C^*$ algebra $Q$ of continuous $[0,1] \rightarrow A$ such that $f(0)\in J$. The map $Q \rightarrow Q$ is $*$-homotopically $f \mapsto c_{f(0)...
0
votes
1answer
22 views

double commutant of the left representation of $C^*$ algebra

If $A$ is a unital $C^*$ algebra with a trace $\tau$,$\lambda:A\to B(L^2(A,\tau))$ is the left representation.where $L^2(A,\tau)$ is the GNS space.Let $\lambda(A)^{"}$ be the double commutant of $\...
4
votes
1answer
62 views

Applying functional calculus to the bounded operator $(T \pm iI)^{-1} $

This is the context: What I wish to prove: 2.3 page 16. Let $T$ be an essentially self-adjoint operator on a Hilbert space $H$. (Now we take its closure) There is a unique homomoprhism of $C^*$ ...
2
votes
1answer
46 views

Apparently a catastrophe: unital injective *-Hom implies factorization of Algebra into tensor product

In a recent lecture we saw the following result, which was hailed as somewhat of a catastrophe: We denote by $\operatorname{M}_m$ be the set of $m\times m$ matrices and by $(\hat A,\hat\varphi)$ a ...
1
vote
1answer
33 views

Injective $*$-homomorphism is isometric

I am aware there are other proofs of line of this statement. But I am interested in the argument outlined here on page 62-63 Corollary II.2.2.9 Let $A$ and $B$ be $C^*$ algebras, $\phi:A \...
1
vote
1answer
14 views

GNS construction I

Let $A$ be a $C^*$ algebra, $\phi$ a positive linear functional on $A$. Put a pre-inner product on $A$ by $\langle x,y \rangle _{\phi} = \phi(y^*x)$. Let $$N_\phi := \{x \in A \, : \, \phi(x^*x) = 0 ...
0
votes
1answer
25 views

embedding $L(H_1) \hookrightarrow L(H_1 \otimes H_2)$

Let $H_1, H_2$ be Hilbert spaces. Let $L(H_i)$ ,$L(H_1 \otimes H_2)$ denote the bounded linear operators on the spaces. There is a canonical map $$L(H_1) \rightarrow L(H_1 \otimes H_2), \quad ...
2
votes
0answers
15 views

Dimension of kernel of closure of an unbounded operator

Let $H$ be a Hilbert space. $$T:D \rightarrow H$$ be a densely defined unbounded operator. Suppose $\dim \ker T < \infty$. Let $\bar{T}$ be its closure, supposing its existence. Is it true ...
2
votes
0answers
38 views

Universal enveloping von Neumann algebra of a separable $C^*$ algebra

Let $A$ be a separable $\mathrm{C}^*$-algebra and let $\pi_U$ be its universal representation. Denote by $M=\pi_U(A)''$ the universal enveloping von Neumann algebra of $A$ (which is isomorphic to $A^{*...
1
vote
0answers
40 views

Index of differential operators are always $0$?

So I got really confused on how one defines the index of an elliptic PDO, after all the background analysis. I have read everything up to page 50 Theorem 3.7.4. I believe this is the core on how one ...
0
votes
0answers
80 views

Decomposition for essentially self adjoint operators.

I am trying to understand page 12 of this notes. In particular Corollary 2.12: If $D$ is a self adjoint operator on a Hilbert space $H$, and if $D$ has compact resolvent, then the kernel of $D$ ...
1
vote
1answer
58 views

Showing a C* algebra with certain properties has a minimal projection

I am trying to show the following which is stated in Exercise 10.11.10 of Blackadars book on K-theory for operator algebras. A unital, simple, nuclear, stably finite, infinite dimensional C*-algebra ...
2
votes
0answers
41 views

Hilbert C*-Modules: Inner *-Isomorphisms

I have got a very basic question, but it would simplify some things, so I hope this resolves in either the affirmative or maybe someone can provide an I guess simple non-example. Given pairs of ...
2
votes
1answer
70 views

Extending $\|H^{\frac{1}{2}}XK^{\frac{1}{2}}\|\leq\frac{1}{2}\|HX+XK\|$ from matrices to operators

I saw in some literature that many author works in finite dimensional (matrix) is because it can be extended into infinite dimensional (operator). The case is as follows: If the following inequality ...
0
votes
1answer
23 views

$KK$-groups definitions

In $K$ Theory of Operator Algebras, page 144 and a paper by Skandalis, page 35 the $KK$ groups are defined differently: Both are triples $(E,\phi, F)$ but Skandalis does not require the condition ...
1
vote
1answer
27 views

Compact Hausdorff space and its double dual: Gelfand Naimark.

I am trying to prove the equivalence of categories between compact Hausdorff spaces and unital $C^*$ algebras. The maps \begin{align*} f: X \mapsto C(X) \\ g: A \mapsto \hat{A} \end{...
0
votes
0answers
19 views

On standard form of particular subalgebra of vN algebra

If $M$ is in standard form, consider the action of finite group on $M$, does the fixed point sub algebra under the action is in standard form?
0
votes
0answers
21 views

Maximal tensor product of quotient C*-algebras

This question is from the book: C*-algebras and Finite-Dimensional Approximations by N.P.Brown and N. Ozawa Ex 13.3.5. Let C$_i$ (i=1,2) be C*-algebras with the LLP and J$_i$ be a closed two-...
2
votes
1answer
33 views

$f+g$ $*$-homomorphsim if and only if $im \, f \cdot im\, g = 0$

Let $f,g:A \rightarrow B$ be $*$-homomorphisms of $C^*$ algebras. Then $f+g$ is a $*$-homomorphism if and only if $im \, f \cdot im \, g =0$. How does this hold? My thoughts: We know that $f+g$ is ...
1
vote
0answers
22 views

K Theory of $C^*$ algebras I, Higson's notes

Let $B$ be a $C^*$ algebra, and $A(B)$ denote the algebra of bounded continuous functions from $[1,\infty)$ to $B$. Let $I(B)$ be the ideal of functions which vanish at infinity. Let $Q(B)$ be the ...
3
votes
0answers
35 views

When is a surjective homomorphism between two unital Banach algebras bounded?

Let $A$ and $B$ be unital Banach algebras and $\theta : A\to B$ a surjective homomorphism between these two spaces. What is a sufficient requirement for $\theta$ being bounded, and how would a proof (...
2
votes
1answer
74 views

What does the statement (Let $K$ be the choquet simplex of all probability measures on $X$) mean?

Let $X$ be a probability measure space. What does the statement (Let $K$ be the choquet simplex of all probability measures on $X$) mean ? It is mentioned in C. Lance. Ergodic Theorems for Convex ...
2
votes
3answers
36 views

Does $\operatorname{Tr}(e^x(\lambda,\infty) x) =\operatorname{Tr}(px)$ imply that $e^x(\lambda,\infty) \le p \le e^x[\lambda,\infty)$?

Let $H$ be a Hilbert space and $\operatorname{Tr}$ be the standard trace on $B(H)$. Let $x$ be a self-adjoint operator in $B(H)$. Let $e = e^x(\lambda,\infty)$ be the spectral projection. Assume that $...