All Questions
Tagged with operator-algebras or oa.operator-algebras
2,153 questions
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On $s$-numbers in finite von Neumann algebra
$T$ is an operator in $M$, $M$ is finite von Neumann algebra. There is a notion of singular value function that is ($s$-numbers). My question is: what is $s$-number for tensor product of two operators ...
- 677
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72
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weakly amenable weighted sequence algebras
Let $v=(v_n)_{n\in\mathbb{N}}$ be a positive weight with $\inf_nv_n>0$ (for convenience we may take $v_n\geqslant1$). Then $\ell_{\infty}(v)$ is a Banach algebra with coordinate-wise multiplication....
- 375
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85
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Injectivity of Fermion algebras
Is the Fermion algebra or $-1$-Fock space as defined in https://arxiv.org/pdf/math/0303045.pdf hyperfinite? Any references?
- 455
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328
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Find the trace for some elements in group algebra
Let $K=\langle b,c,d\mid b^{2}=c^{2}=d^{2}=bcd=1\rangle $. Now we consider $$D=K*\mathbb Z/2\mathbb Z=\left\{a,b,c,d\mid a^{2}=b^{2}=c^{2}=d^{2}=bcd=1\right\}$$ where $*$ is the free product. Then we ...
- 407
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On index 2 and square of subfactors without extra intermediate
Let $N \subsetneq K_i \subsetneq M$, $i=1,2$, be a square of irreducible finite index unital inclusion of hyperfinite ${\rm II}_1$ factors, such that there is no extra intermediate, with $K_1 \not \...
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56
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Sub-matrices with a real spectrum
This question arises from the study of PT-symmetric quantum mechanics.
Let $A$ be an $n\times n$ complex-valued matrix with a real spectrum.
If $A$ is Hermitian, then any sub-matrix corresponding to ...
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145
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Group which is not MF or AF
Does someone know example of group (countable, discrete) which can not be embedded (monomorphism) into
$$ U(\prod M_n/\oplus M_n)$$
unitary group of universal MF-algebra? Or example of group which can ...
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134
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semifinite projection
Let $M$ be von Neumann algebra, $p$ be semiefinite projection and $q$ be projection in $M$ such that $Z(q)=Z(p)$.
( $p$ is semifinite projection if every nonzero subprojection of $p$ contains a ...
- 101
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311
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Chinese remainder theorem for cyclic subfactor planar algebras
This post was inspired by an exchange with the indian woman mathematician Ajit Iqbal Singh.
The chinese remainder theorem can be stated as follows:
Let $n_1, \dots, n_r \ge 2$ be positive integers ...
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76
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The completely reducible bimodules coming from subfactors
This post is a sequel of: Are all the R-R-bimodules completely reducible?
Question: For which (as general as possible) class of subfactors $(N \subset M)$, the bimodule $_NM_M$ is known completely ...
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56
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When is a cycle in $KK^G(A,A)$ with zero operator the identity cycle?
Given a cycle of the form $(\pi,H,0)$ in $KK^G(A,A)$, when is it equivalent to the identity cycle $1_A=(i_A,A,0)$?
The operator $T=0$, and $\pi:A \rightarrow L(H)$ may be injective.
Any criterions ...
- 58
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85
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Is $KK^G(\mathbb{C}^n,B)$ countably additive in $B$ and countable?
Let $G$ be a finite discrete groupoid, $A=\mathbb{C}^n$ a finite dimensional, commutative $C^*$-algebra and assume we have given a $G$-action on $A$. Note that the action of $G$ on $\mathbb{C}^n=C_0(\{...
- 58
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201
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Range of a trace preserving completely positive projection
I'm working in $M_n(\mathbb{C})$, the algebra of complex $n\times n$ matrices. I managed to build a completely positive, trace preserving, star preserving, projection $P$. That is
$$\text{Tr}(P(A)) = ...
- 515
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122
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Is there an irreducible subfactor with an infinite homogeneous single chain lattice?
We know that we can build an irreducible subfactor realizing a finite single chain lattice containing any finite index irreducible maximal subfactors, by using the free composition (see here).
Now ...
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255
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Bounded operators with infinite matrix representations
I asked this question on StackExchange originally, but I'm giving it a go here as well.
Suppose that $A$ is a unital $C^*$-algebra, $\varphi\colon A\to B(H)$ is a unital, completely positive map and ...
- 123
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232
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Morphisms associated to measured spaces [duplicate]
In a previous discussion (von neumann algebras and measurable spaces), the connexion between von Neumann algebras and localized measured spaces was clarified. I would like to have a category theory ...
- 585
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129
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A special Lie subalgebra
Motivated by comments of the following post A question on involutions on the Lie algebra of vector fields we ask the following question:
Let $L$ be a Lie algebra. We consider the Lie subalgebra $$...
- 356
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184
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Can I define Fredholm Index using $\dim \ker ST - \dim \ker TS$?
$X$, $Y$ are Banach spaces.
Let $S \in L(X, Y)$, $T \in L(Y, X)$, where $L(X, Y)$ denotes the Banach algebra of bounded linear operators from $X$ to $Y$. If we have that $Id_Y - ST \in \mathbb{K}(Y)$ ...
- 365
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410
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A noncommutative vector bundle
We know that a noncommutative vector bundle is a finitely generated projective $A$-module where $A$ is a non commutative $C^{*}$ algebra. In this question we introduce a particular non commutative ...
- 356
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293
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Lifting triangles in K-theory to KL-groups
Let $X$ and $Y$ be finite simplicial complexes (or $CW$-complexes) so that $Y\subseteq X$. Let $s\colon C(X)\to C(Y)$ be the map given by restriction. In particular $K_{*}(C(X))$ and $K_{*}(C(Y))$ are ...
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150
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A question about multiplier algebra of $C_0(G)\otimes C_b(G)$ for a locally compact group $G$
Let $G$ be locally compact group. How we can show that
$$
M(C_0(G)\otimes C_b(G))=C_b(G,C_b(G)).
$$
($M(C_0(G)\otimes C_b(G))$ is the multiplier algebra of $C_0(G)\otimes C_b(G)$)
- 1
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152
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Need help determining whether a certain map is a $C^\ast$ homomorphism
Hello, I need help determining whether the map I defined between two algebras is a well-defined homomorphism of $C^\ast$-algebras. I ran into this problem while trying to define a "rotation map" ...
- 365
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200
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Weights on Von Neuman factors
Let A be a type $I$ factor on a Hilbert space H. Let $\varphi$ be a semi-finite normal weight on $A^{+}$ is it possible to say that then there exist Hilbert spaces $H_2 \subset H_1$ and an isomorphism ...
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144
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isomophism, commutator
Let X be a Banach space. $B(X)$ is the algebra of all bounded linear operators on X.
$\phi: B(X)\rightarrow B(X)$ is a isomoprphisn, $\varphi: B(X)\rightarrow B(X)$ is a isomorphism or negative anti-...
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354
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abstract algebra for component wise operations on "vectors" or what it might be called
I have a quite tough problem to solve and need an algebra that allows to "vectors" following operations:
- multiplication between two vectors are componentwise that means v=(v1, v2, v3,...) multiplied ...
- 254
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194
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What methods have been used to study AW*-algebras up to now?
I am interested mainly in ring theory and homological algebra. Now I want to know about the research methods of AW*-algebras. So I want to know the answer to the question:"what methods have been used ...
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218
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Series of linear maps: on a paper by Evans and Hanche-Olsen
I was reading this paper by Evans and Hanche-Olsen. In theorem 2, there are six equivalent statements given. I write just two of them, which I want to use.
Let $L$ be a bounded self-adjoint
...
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164
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Can we separate Toeplitz matrices for negative and positive eigenvalues?
Consider a Toeplitz matrix T which has both positive and negative eigenvalues. Can we prove that there exist two Toeplitz matrix T1 and T2 such that T1+T2=T and T1 has only one positive Eigenvalues ...
- 9
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185
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Characterization of Complex Group Algebras
Is there a way to characterize which complex algebras arise as the group algebra of some locally compact group? To make this more concrete, say $A$ is a sub-algebra of $\text{Mat}(n,\mathbb{C})$, $n\...
- 445
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416
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norm one approximate identities in separable C* algebras
I'm trying to prove Corollary 1.4.9 in K. Davidson's book (Exercise 1.5):
If A is a separable C* algebra, then there is an increasing sequence
$E_i, i=1,...,\infty$ of positive norm-one elements ...
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167
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Primitive Ideals of $\mathcal{B}_0(\mathcal{H})^\sim$ (identity adjoined)
A note I'm studying says its primitive ideals space is $\lbrace \lbrace 0 \rbrace, \mathcal{B}_0(\mathcal{H}) \rbrace $. I think it might be just $\lbrace \lbrace 0 \rbrace \rbrace $ so I'm somewhat ...
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1k
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Linear Mapping and integration
I have been reading the paper - "Introduction to Quantum Fisher Information".
In section 1.2 the author talks about the linear map $\mathbb{J}_D$, which he defines as follows:
Let $D \in M_n$ be a ...
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320
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A result about Fredholm operator
When I read the article "Index Theory" in Handbook of global analysis, I meet a result as below(Corollary 2.13):
If every $F_0\in \mathcal {F}(H_1,H_2)$, there is an open neighborhood $U_0\subseteq \...
- 381
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2
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158
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Recover hermitian matrices out of the knowledge of their trace
Let $M_{1},\ldots, M_{m}$ be $m$ hermitian $N\times N$ matrices, and $tr=\frac{1}{N}Tr$ the normalised trace on the algebras of such matrices. If you know the quantities $tr(M_{i_{1}}\ldots M_{i_{...
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382
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derivation between two $C^{*}$ algebras
given two $C^{*}$ algebras $A\subset B $, acting on the same Hilbert space $H$, and $\delta $ is a derivation from $A$ into $B$,(in this case, a derivation is a linear mapping such that $\delta(ab)=a\...
- 1,528
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2
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640
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Invariance of spectrum under conjugation
Let $T$ be a self-adjoint invertible operator on $\mathcal{H}$ with a continuous spectrum, means the spectral measure is nonatomic. For which class of invertible operators $V$( with continuous ...
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1
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102
views
Is an $A$-$B$—$C^*$-correspondence a representation of a $G$-$C^*$-algebra, $\rho \colon A \otimes_{ \alpha } B \to \mathcal{L} ( \mathcal{H} )$?
Let $R$ and $S$ be two rings.
It is known that an $R$-$S$-bimodule is actually the same thing as a left module over the ring $R \otimes_{\mathbb{Z}} S^{\mathrm{op}}$, where $S^{\mathrm{op}}$ is the ...
- 595
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246
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Density of normal elements in a C*- algebra [closed]
Let $A$ be a unital C*-algebra.
I wanted to know if there is a necessary and sufficient condition for normal elements to be dense in $A$?
- 149
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180
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On spectral multiplicity of left shift operators
Let $U$ be an operator defined on $l^{2}(\mathbb{Z})$ by $U(e_{n})=e_{n-1}$, where $e_{n}$ is an orthonormal basis of $l^{2}(\mathbb{Z})$. $U$ is a left shift operator. Since $U$ is unitary operator ...
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164
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Closure of the point spectrum of an unbounded diagonalizable operator
Given a (separable) Hilbert space H and an unbounded densely defined linear operator $T:{\cal D}(T) \to $H such that ${\cal D}$ is diagonalizable (it means $\exists$ an O.N.B. of H such that all basis ...
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210
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A commuting pair of isometries
Let $H$ be a Hilbert space and $B(H)$ be the space of all bounded operators on $H$.
The Wold decomposition says that: an operator $x$ in $B(H)$ is an isometry if and only if $x=x_u\oplus x_s$ where $...
- 4,058
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1
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90
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Strictly increasing approximation of the identiy
Is there always a strictly increasing approximation of the identity in a separable $C^*$-algebra?
- 49
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230
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Determine whether the center of a $C^*$-algebra is 0
Let $G$ be a locally compact group and $A$ be a non-unital $C^*$-algebra. )$(A,G,\alpha)$ is a $C^*$-dynamical sysytem. The space of all continous functions from $G$ to $A$ with compact support is ...
- 451
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784
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What is a type $\text{II}_1$ factor von Neumann algebra?
After finding formal definitions in various texts (see, eg, Witten, Notes On Some Entanglement Properties Of Quantum Field Theory, Rev. Mod. Phys. 90, 45003 (2018), doi:10.1103/RevModPhys.90.045003 ...
- 227
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1
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132
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How should $A^α$ be defined for real $α ∈ [0,∞)$ and $A\in M_n(\mathbb C)$? [closed]
Let $A\in M_n(\mathbb C)$ be arbitrary. I'm interested to know How should $A^{\alpha}$ be defined for real $\alpha\in [0,\infty)$? When $A$ is nonsingular, we can define $A^{\alpha}=\exp(\alpha \log(A)...
- 201
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1
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143
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Relationship between noncommutative torus for different values of theta [closed]
Let $u,v\in B(L_2(\mathbb T))$ defined as $u(f)(z)=zf(z)$ and $v(f)(z)=f(ze^{-2\pi i\theta})$ for $z\in\mathbb T$ where $\theta\in\mathbb R\setminus\mathbb{Q}$. Denote the $C^*$ algebra generated by $...
- 1,879
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1
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138
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Weak center is same as center for $C^{\ast}$-Algebra? [closed]
Let $A$ be a $C^{\ast}$-algebra. We say $A$ is weakly commutative if $ab^*c=cb^*a$ for all $a,b,c \in A$ and define weak center of $A$ as $$Z_w(A)= \{ v \in A : av^*c=cv^*a \;\forall a,c \in A \}.$$
...
- 1,115
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1
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217
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The norm of the difference of two normal states
Let $M$ be a type III$_1$ factor and $\rho$ be a normal state on $M$. If $p$ is a projection in $M$, can we find another normal state $\rho'$ on $M$ such that $\rho'(p)=0$ and $\|\rho-\rho'\|=k\rho(...
- 427
-2
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0
answers
41
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Do the domains of the two square roots of a positive (unbounded) operator coincide? [closed]
Let $H$ be a Hilbert space and $D:\mathrm{Dom}(D) \to H$ a densely defined operator on $H$. We further assume that $D$ is closed and self-adjoint. If we further assume that $D$ is positive, then we ...
- 417
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1
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325
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Is the category of $Z^* $ algebra equivalent to the category of $C^*$ algebras
A $Z^*$ algebra is a $C^*$ algebra whose all positive elements are zero divisor.
They form a category with usual structures.
Question. Is this category equivalent to the category of $C^*$ algebras?
...
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