Algebras of operators on Hilbert space, $C^*-$algebras, von Neumann algebras, non-commutative geometry

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Unitizations of Banach algebras and matrix norms

Consider the short exact sequence of Banach algebras $0\rightarrow A\rightarrow A^+\rightarrow\mathbb{C}\rightarrow 0$ where $A$ is a Banach algebra without unit and $A^+$ denotes the unitization of ...
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42 views

Relationship between group $C^\ast$-algebras $C^\ast(G)$ and graph $C^\ast$-algebras $C^\ast(E)$

Let $E=E(G,S)$ be the graph defined by a group $G$ and a subset $S$ of $G$. What is relationship between group $C^\ast$-algebras $C^\ast(G)$ and graph $C^\ast$-algebras $C^\ast(E)$?
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43 views

Can a semigroup be defined on a Banach algebra? [on hold]

I simply need to know that how a semigroup of operators (say $\{T(t)\}_{t\geq 0}$) is defined on any Banach algebra (say $X$)? For $(f,g\in X)$ now the so called product is also there i.e. $f.g\in X$. ...
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1answer
52 views

A relation among projections of a von Neumann algebra

This is a follow-up question on this. Let $A$ be a von Neumann algebra and $P$ be its projection lattice. For $p,s,q \in P$, let us define $ p \perp q \mid s \iff ps^\perp q = 0$ where $s^\perp = ...
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How “nondegenerate” are amalgamated free products of C*-algebras?

In the following, I assume all algebras are unital. Let $A$ and $B$ be C*-algebras that each contain (isomorphic copies of) a common C*-subalgebra $C$. Let $A *_C B$ denote the amalgamated free ...
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1answer
110 views

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 ...
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180 views

Can $C^*$-algebra of continuous functions on $R^n$ ($S^n$) be characterized alternatively?

Dictionary between algebra and geometry is somewhat one of the main concepts in modern mathematics. So commutative $C^*$ algebras are one-to-one with locally compact Hausdorff spaces. So it is ...
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37 views

Support vectors and relative modular operator

I'm studying the relative modular operator and I'm looking for o good text to do it. Until now I'm using Araki's papers but I don't know how to deal with the support of a vector, $s^M(\xi)$, which is ...
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128 views

$K$-Theory of finite dimensional Banach algebras

Is there a finite dimensional Banach algebra $A$ for which $K_{0}(A)$ is a finite group? I asked this question in MSE but I received no answer ...
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104 views

References for a lemma about compact operators on a Hilbert module

I am looking for a reference for the following result: If $A$ and $B$ are C* algebras, $H$ is a right Hilbert $A$-modules, $\phi :A \rightarrow B$ is a morphism, and assume that there is a map $\eta ...
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342 views

Who gave the generalized Stone-Weierstrass Theorem?

Let $X$ be a compact Hausdorff space and $\mathcal{A}$ be a closed self-adjoint subalgebra of $C(X)$ which contains the constants. Then $\mathcal{A}$ is the collection of continuous functions on ...
6
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1answer
113 views

Spectral mapping theorem for polynomials in $z,\overline{z}$ and direct construction of the function calculus for a normal operator

Suppose that $A$ is an element in Banach algebra and $p$ is a polynomial. Then we have an equality $p(\sigma(A))=\sigma(p(A))$ where $p(A)$ has an elementary meaning. This theorem (the spectral ...
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363 views

States in C*-algebras and their origin in physics?

in $C^*-$algebras with unit element, there is the definition of a state, as a functional $\omega$ with $\omega(e)=||\omega||=1.$ Now, of course there is also in classical physics and quantum ...
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126 views

Weak convergence implies norm convergence for trace class operators?

It is known that weak convergence implies norm convergence in $\ell^1(\mathbb{N})$, see e.g. here. Because of the typical analogies of the Schatten ideals $C_p \subset B(H)$ (where $H$ is a Hilbert ...
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1answer
123 views

The positive cone of the standard representation of a Von Neumann algebra

Let $A$ be a von Neumann algebra, let $L^2(A)$ be the underlying Hilbert space of the standard form of $A$, and $P \subset L^2(A)$ the canonical positive cone (see for example this paper by Haagerup). ...
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213 views

Is there a link between $H_2(G,\mathbb{Z})$, the Schur Multiplier of a group, and the “other” Schur multipliers of a group?

The name for the the following 2 mathematical objects: $$H_2(G,\mathbb{Z})$$ and $$\{K:G\times G\longrightarrow\mathbb{C}\ |\ \forall T\in B(l^2(G))\text{we have that}~S:G\times ...
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122 views

Submodules of a Hilbert space with finite Jones index with respect to a von Neumann algebra

While studying some basic theory of Cartan subalgebras of von Neumann algebras I found the following fact that I couldn't prove: Let $H$ be a Hilbert space, $A$ and $B$ trace von Neumann subalgebras ...
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97 views

The left regular representation of the Jacobi groups over local fields of characteristic >2 is type I?

Let $K$ be a non-archimedean local field of characteristic $>2$. Consider the Jacobi group $G=H_{2n+1}(K)\rtimes Sp_{2n}(K)$, which is the semidirect product of the Heisenberg group $H_{2n+1}(K)$ ...
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144 views

Uniqueness of the reduced free product of unital completely positive maps

For $1\leq i\leq n$, let $\psi_i$ be a faithful state on the C$^*$-algebra $A_i$ and $\phi_i$ be a faithful state on the C$^*$-algebra $B_i$. Let $(A,\psi) = *_{i=1}^n (A_i,\psi_i)$ and $(B, \phi) = ...
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1answer
161 views

Polar decomposition

Let $x$ be a trace class operator on a Hilbert space $H$. Then $x$ induces unique normal functional on $B(H)$, which we denote it by $f_x$. Let us consider the polar decomposition $x=u|x|$ and ...
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1answer
78 views

Commutator representation of certain smoothing operators

I have a question regarding the classical trace $\text{Tr} \colon \Psi^{-\infty}(S^1)\to \mathbb C$ on pseudodifferential operators of infinite negative order (i.e. smoothing operators), defined over ...
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Is the generated subalgebra of a subset of pairwise operator-commuting element in a JB-algebra associative?

In a Jordan algebra elements $a$ and $b$ are said to operator-commute, whenever $a \circ (b \circ x) = b \circ (a \circ x)$ for every other element $x$. (That is: $T_aT_b = T_bT_a$, writing $T_x(y) = ...
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112 views

Functoriality of $\mathsf{Cu}$

I have always been happy with the proof of the functoriality of the Cuntz semigroup $\mathsf{Cu}$ given in arXiv:0902.3381, where the isomorphism $$\mathsf{Cu}(A)\cong W(A\otimes K)$$ is used, $A$ ...
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1answer
199 views

$C^{*}$-correspondences viewed as generalized endomorphisms

I've heard that $C^{*}$-correspondences (over a $C^{*}$-algebra) can be viewed as generalized endomorphisms of the algebra. I would like to understand this, and be pointed towards books or papers ...
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78 views

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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1answer
124 views

How do we know the map is $w^{*}$-continuous?

I am reading a paper by David Blecher, which contains the following: " If $T: Y \to Z$ is a surjective isometric module map between $W^{*}$-modules over $M$, then $T$ is unitary. Also, $T$ is a ...
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107 views

Integral representation of the complex homogeneous polynomial $z_1\cdots z_n$

Consider the transform (see e.g., (5.1) in this paper): \begin{equation*} \Lambda_\mu(q)(z) := \int_{\Delta_n} ...
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76 views

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 ...
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173 views

An inequality for Fuchsian groups?

Let $G$ be a finitely generated Fuchsian group. (i.e. a discrete subgroup of $\mathrm{PSL}_2(\mathbb{R})$). Is it true that $d(G) < 2\beta_{2}^1(G) + 1$ ? Here, $\beta_{2}^1(G)$ stands ...
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1answer
82 views

Approximation of the central support

Let $(M,\tau)$ be a tracial von Neumann algebra, i.e. a unital subalgebra $M=M''\subset \mathbb{B}(H)$; a finite (faithful) trace $\tau: M\to \mathbb{C}$ (faithful means that $\tau(x^*x)=0$ ...
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Commutation preserving operators

Let $A$ and $B$ be unital $C$*-algebras and let $T\colon A\to B$ be a bounded linear bijection that preserves commuting elements, i.e., $ab=ba$ implies $TaTb=TbTa$. Does $T^{**}$ then also preserve ...
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150 views

States and left ideals

Given a nontrivial left ideal $J$ of a unital $C^*$ algebra $A$, is there a state on $A$ which vanishes on all elements of $J$? (Left or right doesn't matter, just not 2-sided.) The problem came ...
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136 views

Connectivity of the group of invertible elements of $C(S^{2})\otimes A$

For what type of $C^{*}$ algebras $A$, the group of invertible elements of $C(S^{2}) \otimes A$ is a connected group? All finite dimensional $A$ satisfy this property. Is it true to say ...
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99 views

Comparison between spectra

Let $G$ be a normal operator with compact resolvent on a Hilbert space $H$ such that ${\rm ker}(G) \neq {0}$. Further let $P$ be the orthogonal projection onto ${\rm ker}(G)$, and let $G_{0}:=G+P$. ...
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Extending Akemann's Non-Commutative Urysohn Lemma

Assume $A$ is a C*-algebra and $p,q\in A^{**}$ are compact projections. Can we always find $a,b\in A^1_+$ with $p\leq a$, $q\leq b$ and $||pq||=||ab||$? Note if $||pq||=1$ this is immediate, ...
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219 views

A question on complex line bundle over $S^{2}$

Consider the trivial bundle $\epsilon_{2}=S^{2}\times \mathbb{C}^{2}$ with the standard Hermitian inner product $<(a,b), (c,d)>=a\bar{c}+b\bar{d}$. Assume that $\ell$ is a sub line bundle of ...
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Almost idempotent approximate units in C*-algebras

As in Blackadar's "Operator Algebras" Definition II.4.1.1., call an approximate unit $(a_\lambda)$ in the positive unit ball of a C*-algebra almost idempotent if $a_\lambda a_\gamma=a_\lambda$ ...
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162 views

Discrete groups G whose full C*-algebra C*(G) is not MF?

This is a cheap rip-off of this question, but I am genuinely interested in an answer. Is there a known example of a countable discrete group G whose full group C*-algebra C*(G) is not MF? Let us ...
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112 views

Closed containment of open projections in C*-algebras

For a C*-algebra $A$ and open projections $p,q\in A^{**}$, consider the following statements. $\overline{p}\leq q$ $p\leq q$ and there exists open $r\in A^{**}$ with $rp=0$ and $r\vee q=1$ $p\leq q$ ...
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$H^{*}$ algebras as a generalization of $C^{*}$ algebras

Let $H$ be the quaternions algebra. An $H^{*}$ algebra is a normed ring $A$ which is simultaneously a unital left $H$ module and has an involution $*$ with the following properties: $\forall \lambda ...
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simple and non nuclear $C^*$-algebra

Is there an example of simple and non-nuclear(non-amenable) $C^*$-algebra?
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On the tensor product of irreducible finite index subfactors

This post extends this question on maximal subfactors, admitting this answer of Feng Xu. Statement: A tensor product of irreducible finite index subfactors has a non-obvious intermediate if and only ...
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Graded structures for simple $C^{*}$ algebras without nontrivial idempotent

Edit(A confession): I just realized that the question is trivial: Since one can easily prove that the convex hull of the spectrum of every nontrivial homogeneous element of a $\mathbb{Z}_{n}$-graded ...
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256 views

Are free positive operators equivalent to almost-commuting operators?

Set $A:=C_0((0,1]) * C_0((0,1])$ (the free product C*-algebra), with canonical generators $a,b$ (positive contractions). Does there exists some $\gamma>0$ such that, for any $x,y \in A$ if $x^*x=a$ ...
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second dual of minimal tensor products of $C^*$-algebras

Let $A$ be a unital $C^*$-algebras and $K(H)$ is $C^*$-algebras of compact operators on separable Hilbert space $H$. Is it true that $(A \otimes K(H))^{**}= A^{**} \overline{\otimes}B(H)$?
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187 views

Fundamental class in $KO[1/2]$

Let $M^m$ be an oriented Riemannian manifold. The signature operator associates to $M$ a class $\Delta_M\in KO_m(M)[1/2]$. I have two questions about this class $\Delta_M$: Rationally, $\Delta_M$ is ...
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1answer
190 views

Power's Theorem for irreducible representations

Let $A_{\alpha}\subset B(H)$ be a bunch of unital C*-algebras acting on a Hilbert space $H$ given together with their character spaces $M(A_{\alpha})$'s. A very nice theorem of Stephen C. Power ...
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174 views

Groups without property (T) but all finite quotients are expanders

What is an example of a group $G$ which 1- is finitely generated by $S$, 2- does not have property (T), 3- admits infinitely many finite quotients which do not factor through an homomorphism $G ...
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180 views

Equivariant Fredholm operators classify equivariant K-theory

Let $\mathcal{F}$ be the space of Fredholm operators on a separable Hilbert space $H$ with the topology induced by the operator norm. If $X$ is compact, Atiyah-Jänich proved that ...
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88 views

Existence of free operators, independent and with given distributions

Excuse me if the question is not appropriate for Mathoverflow. I havs asked it in math.stackexchange, but did not get any response. And so, I dared to put it here. I am trying to learn free ...